![viii Contents
3.5. Computing the Dimension of a Commutative Subalgebra 114
Biographical Note on Frobenius 123
4. The Module Setting 124
4.1. A Modicum of Modules 126
4.2. Direct Sum Decompositions 135
4.3. Free and Projective Modules 144
4.4. Von Neumann Regularity 152
4.5. Computing Quasi-Inverses 159
4.6. The Jordan Form Derived Module-Theoretically 169
4.7. The Weyr Form of a Nilpotent Endomorphism:
Philosophy 174
4.8. The Weyr Form of a Nilpotent Endomorphism: Existence 178
4.9. A Smaller Universe for the Jordan Form? 185
4.10. Nilpotent Elements with Regular Powers 188
4.11. A Regular Nilpotent Element with a Bad Power 195
Biographical Note on Von Neumann 197
PARTTWO:Applications of the Weyr Form 199
5. Gerstenhaber’s Theorem 201
5.1. k-Generated Subalgebras and Nilpotent Reduction 203
5.2. The Generalized Cayley–Hamilton Equation 210
5.3. Proof of Gerstenhaber’s Theorem 216
5.4. Maximal Commutative Subalgebras 221
5.5. Pullbacks and 3-Generated Commutative Subalgebras 226
Biographical Notes on Cayley and Hamilton 236
6. Approximate Simultaneous Diagonalization 238
6.1. The Phylogenetic Connection 241
6.2. Basic Results on ASD Matrices 249
6.3. The Subalgebra Generated by ASD Matrices 255
6.4. Reduction to the Nilpotent Case 258
6.5. Splittings Induced by Epsilon Perturbations 260
6.6. The Centralizer of ASD Matrices 265
6.7. A Nice 2-Correctable Perturbation 268
6.8. The Motzkin–Taussky Theorem 271
6.9. Commuting Triples Involving a 2-Regular Matrix 276
6.10. The 2-Regular Nonhomogeneous Case 287
6.11. Bounds on dim C[A1, . . . , Ak] 297
6.12. ASD for Commuting Triples of Low Order Matrices 301
Biographical Notes on Motzkin and Taussky 307](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-13-320.jpg)
![Preface xiii
Weyr form, for which the centralizer description is simpler. It is this
property that gives the Weyr form an edge over its Jordan counterpart
in a number of applications.
4: The Module Setting
The Jordan form has a known ring-theoretic derivation through
decompositions of finitely generated modules over principal ideal
domains. In this chapter we derive the Weyr form ring theoretically,
but in an entirely different way, by using ideas from decompositions
of projective modules over von Neumann regular rings. The results
suggest that the Weyr form lives in a somewhat bigger universe than its
Jordan counterpart, and is perhaps more natural.
PART II: APPLICATIONS OF THE WEYR FORM
5: Gerstenhaber’s Theorem
The theorem states that the subalgebra F[A, B] generated by two
commuting n × n matrices A and B over a field F has dimension at
most n. It was first proved using algebraic geometry, but later Barría
and Halmos, and Laffey and Lazarus, gave proofs using only linear
algebra and the Jordan form. Here we simplify the Barría–Halmos
proof even further through the use of the Weyr form in tandem with the
Jordan form, utilizing an earlier duality.
6: Approximate Simultaneous Diagonalization
Complex n × n matrices A1, A2, . . . , Ak are called approximately
simultaneously diagonalizable (ASD) if they can be perturbed
to simultaneously diagonalizable matrices B1, B2, . . . , Bk. In this
chapter we attempt to show how the Weyr form is a promising tool
(more so than the Jordan form) for establishing ASD of various
classes of commuting matrices using explicit perturbations. The ASD
property has been used in the study of phylogenetic invariants in
biomathematics, and multivariate interpolation.
7: Algebraic Varieties
Here we give a largely self-contained account of the algebraic geometry
connection to the linear algebra problems studied in Chapters 5
and 6. In particular, we cover most of the known results on the
irreducibility of the variety C(k, n) of all k-tuples of commuting
complex n × n matrices. The Weyr form is used to simplify some
earlier arguments. Irreducibility of C(k, n) is surprisingly equivalent
to all k commuting complex n × n matrices having the ASD property
described in Chapter 6. But a number of ASD results are known only
through algebraic geometry. Some of this work is quite recent (2010).](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-18-320.jpg)
![xiv Preface
Our choice of the title Advanced Topics in Linear Algebra indicates that we are
assuming our reader has a solid background in undergraduate linear algebra (see
the introduction to Chapter 1 for details on this). However, it is probably fair
to say that our treatment is at the higher end of “advanced” but without being
comprehensive, compared say with Roman’s excellent text Advanced Linear
Algebra,1
in the number of topics covered. For instance, while some books on
advanced linear algebra might take the development of the Jordan form as one
of their goals, we assume our readers have already encountered the Jordan form
(although we remind readers of its properties in Chapter 1). On the other
hand, we do not assume our reader is a specialist in linear algebra. The book is
designed to be read in its entirety if one wishes (there is a continuous thread),
but equally, after a reader has assimilated Chapters 2 and 3, each of the four
chapters that follow Chapter 3 can be read in isolation, depending on one’s
“pure” or “applied” leanings.
At the end of each chapter, we give brief biographical sketches of one or two
of the principal architects of our subject. It is easy to forget that mathematics has
been, and continues to be, developed by real people, each generation building
on the work of the previous—not tearing it down to start again, as happens
in many other disciplines. These sketches have been compiled from various
sources, but in particular from the MacTutor History of Mathematics web
site of the University of St. Andrews, Scotland [http://www-history.mcs.st-
andrews.ac.uk/Biographies],andI.Kleiner’sAHistoryofAbstractAlgebra.Note,
however, that we have given biographies only for mathematicians who are no
longer alive.
When we set out to write this book, we were not thinking of it as a text
for a course, but rather as a reference source for teachers and researchers. But
the more we got into the project, the more apparent it became that parts of
the book would be suitable for graduate mathematics courses (or fourth-year
honorsundergraduatecourses,inthecaseofthebetterantipodeanuniversities).
True, we have not included exercises (apart from a handful of test questions),
but the nature of the material is such that an instructor would find it rather easy
(and even fun) to make up a wide range of exercises to suit a tailored course. As
to the types of course, a number spring to mind:
(1) A second-semester course following on from a first-year graduate
course in linear algebra, covering parts of Chapters 1, 2, 3, and 6.
1. Apart from our background in Chapter 1, there is no overlap in the topics covered in our book
and that of Roman.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-19-320.jpg)
![xviii Our Style
We make no apology for the use of whimsy.1
In our view, there is a place
for whimsy even within the erudite writings of mathematicians. It can help
put a human face on the authors (we are not high priests) and can energize
a reader to continue during the steeper climbs. Our whimsical comments are
mostly reserved for an occasional footnote. But footnotes, being footnotes, can
be skipped without loss of continuity to the story.
We have tried to avoid the formality of article writing in referencing works.
Thus, rather than say “ see Proposition 4.8 (2) and the Corollary on p. 222
of [BAII] ” we would tend to say simply “ see Chapter 4 of Jacobson’s Basic
Algebra II.” Likewise, an individual paper by Joe Blog that is listed in our
bibliography will usually be referred to as “the 2003 paper by Blog,” if there is
only one such paper. The interested reader can then consult the source for more
detail.
What constitutes “correct grammar” has been a source of much discussion
and ribbing among the three authors, prompted by their different education
backgrounds (New Zealand, Scotland, and U.S.A.). By and large, the British
Commonwealth has won out in the debate. But we are conscious, for example,
of the difference in the British and American use of “that versus which,”2
and in
punctuation. So please bear with us.
Our notation and terminology are fairly standard. In particular, we don’t
put brackets around the argument in the dimension dim V of a vector space
V or the rank of a matrix A, rank A. However, we do use brackets if both
the mathematical operator and argument are in the same case. Thus, we write
ker(b) and im(p) for the kernel and image of module homomorphisms b and
p, rather than the visually off-putting ker b and im p. Undoubtedly, there will
be some inconsistencies in our implementation of this policy. An index entry
such as Joe Blog’s theorem, 247, 256, 281 indicates that the principal statement
or discussion of the theorem can be found on page 256, the one in boldface.
This is done sparingly, reserved for the most important concepts, definitions, or
results. Very occasionally, an entry may have more than one boldfaced page to
indicate the most important, but separate, treatments of a topic.
Finally, a word to a reader who perceives some “cheerleading” on our part
when discussing the Weyr form. We have attempted to be even-handed in our
1. In a 2009 interview (by Geraldine Doogue, Australian ABC radio), Michael Palin (of Monty
Python and travel documentary fame, and widely acclaimed as a master of whimsy) was asked
why the British use whimsy much more so than Americans. His reply, in part, was that Britain
has had a more settled recent history. America has been more troubled by wars and civil rights.
Against this backdrop, Americans have tended to take things more seriously than the British.
2. Our rule is “that” introduces a defining clause, whereas “which” introduces a nondefining
clause.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-23-320.jpg)
![Background Linear Algebra 5
Thatisnottosayoneviewoflinearalgebraisrightandanotherisflatlywrong.
If there are particular points of view that come across in the present book,
their origins probably lie in the time the authors have worked with algebraic
structures (such as semigroups, groups, rings, and associative algebras), and
with having developed a healthy respect for such disciplines as category theory,
universal algebra, and algebraic geometry. Of course, we have also come to
admire the beautiful concepts in analysis and topology, some of which we use
in Chapters 6 and 7. In this respect, our philosophical view is mainstream—
mathematics, of all disciplines, should never be compartmentalized.
The letter F will denote a field, usually algebraically closed (such as the
complex field C), that is, every polynomial over F of positive degree has roots
in F. The space of all n-tuples (which we usually write as column vectors) of
elements from F is denoted by Fn
. This space is the model for all n-dimensional
vector spaces over F, because every n-dimensional space is isomorphic to Fn
. By
the standard basis for Fn
we mean the basis
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1
0
0
.
.
.
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0
1
0
.
.
.
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0
0
1
.
.
.
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, · · · ,
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0
0
0
.
.
.
1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎫
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎭
.
The ring of polynomials, in the indeterminate x and with coefficients from F,
is denoted by F[x]. This ring plays a similar role in linear algebra to that of the
ring Z of integers in group theory. (Both rings are Euclidean domains and, for
instance, the order of a group element, as an element of the ring Z, translates
to the minimal polynomial of a matrix, as an element of the ring F[x].) If V is
a vector space over F (usually finite-dimensional), its dimension is denoted by
dim V. The subspace of V spanned (or generated) by vectors v1, v2, . . . , vn is
denoted by v1, v2, . . . , vn. If U1, U2, . . . , Uk are subspaces of V, their sum is
the subspace
U1 + U2 + · · · + Uk = {v ∈ V : v = u1 + u2 + · · · + uk for some ui ∈ Ui}.
For a linear transformation T : V → W from one vector space V to another
W, the rank of T is the dimension of the image T(V), and the nullity of T is
the dimension of the null space or kernel, ker T = {v ∈ V : T(v) = 0}. We
have the fundamental rank, nullity connection:
rank T + nullity T = dim V.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-32-320.jpg)
![Background Linear Algebra 7
For a polynomial f (x) = a0 + a1x + · · · + amxm
∈ F[x] and square matrix
A ∈ Mn(F), we can form the matrix polynomial
f (A) = a0I + a1A + a2A2
+ · · · + amAm
.
This polynomial evaluation map f → f (A) (for a fixed A), from F[x] to
Mn(F), is a simple but useful algebra homomorphism (i.e., a linear mapping
that preserves multiplication). The Cayley–Hamilton theorem says that over
any field F, every square matrix A vanishes at its characteristic polynomial:
p(A) = 0 where p is the characteristic polynomial of A.
The square matrices A that have 0 as an eigenvalue are the singular
(noninvertible) matrices, because they are the matrices for which the
homogeneous system Ax = 0 has a nonzero solution. A square matrix A is
called nilpotent if Ar
= 0 for some r ∈ N and in this case the least such r is the
(nilpotency) index of A. If A is nilpotent then 0 is its only eigenvalue. (For if
Ar
= 0 and Ax = λx for some nonzero x, then 0 = Ar
x = λr
x, which implies
λ = 0.) Over an algebraically closed field, the converse also holds, as we show
in our first proposition:
Proposition 1.1.1
Over an algebraically closed field, an n × n matrix A is nilpotent if and only if 0 is the
only eigenvalue of A. Also, a square matrix that does not have two distinct eigenvalues
must be the sum of a scalar matrix λI and a nilpotent matrix.
Proof
The second statement follows from the first because if λ is the only eigenvalue of
A, then 0 is the only eigenvalue of A − λI (and A = λI + (A − λI) ). Suppose 0
is the only eigenvalue of A. Since the field is algebraically closed, the characteristic
polynomial of A must be p(x) = xn. By the Cayley–Hamilton theorem, 0 =
p(A) = An and so A is nilpotent.
Note that the argument breaks down (not the Cayley–Hamilton theorem,
which holds over any field) if the characteristic polynomial doesn’t factor
completely. For instance, over the real field R, the matrix
A =
⎡
⎣
0 0 0
0 0 −1
0 1 0
⎤
⎦
has zero as its only eigenvalue but is not nilpotent. (However A3
+ A = 0,
consistent with the Cayley–Hamilton theorem.)
Now Mn(F) is not just a vector space under matrix addition and scalar
multiplication, but also a ring with identity under matrix addition and](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-34-320.jpg)
![8 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A
multiplication, with scalar multiplication and matrix multiplication nicely
intertwined by the law4
λ(AB) = (λA)B = A(λB) for all λ ∈ F and all
A, B ∈ Mn(F). In this context, we refer to Mn(F) as an (associative) algebra
over the field F. (The general definition of an “algebra over a commutative
ring” is given in Definition 4.1.8.) By a subalgebra of Mn(F) we mean a
subset B ⊆ Mn(F) that contains the identity matrix and is closed under scalar
multiplication, matrix addition, and matrix multiplication (in other words,
a subspace that is also a subring). Given a subset S ⊆ Mn(F), there is a
unique smallest subalgebra of Mn(F) containing S, namely, the intersection
of all subalgebras containing S. This is called the subalgebra generated by
S, and is denoted by F[S]. In the case where S = {A1, A2, . . . , Ak} consists
of a finite number k of matrices, we say that F[S] is k-generated (as an
algebra) and write F[S] = F[A1, A2, . . . , Ak]. For a single matrix A ∈ Mn(F),
clearly
F[A] = {f (A) : f ∈ F[x]}.
In fact {I, A, A2
, . . . , Am−1
} is a vector space basis for F[A] if Am
is the first
power that is linearly dependent on the earlier powers. Describing the members
of F[A1, A2, . . . , Ak] when k 1, or even computing the dimension of this
subalgebra, is in general an exceedingly difficult problem.
Over an algebraically closed field F, a nonderogatory matrix is a square
matrix A ∈ Mn(F), all of whose eigenspaces are 1-dimensional. This is not the
same thing as A having n distinct eigenvalues, although the latter would certainly
be sufficient. Nonderogatory matrices can be characterized in a number of ways,
two of which are recorded in the next proposition. We postpone its proof until
Proposition 3.2.4, by which time we will have collected enough ammunition to
deal with it quickly.
Proposition 1.1.2
The following are equivalent for an n × n matrix A over an algebraically closed
field F:
(1) A is nonderogatory.
(2) dim F[A] = n.
(3) The only matrices that commute with A are polynomials in A.
Nowadays, the term “nonderogatory” often goes under the name 1-regular.
The reason for this is that nonderogatory is the k = 1 case of a k-regular
4. This is equivalent to saying that left multiplying matrices by a fixed matrix A, and right
multiplying matrices by a fixed matrix B, are linear transformations of Mn(F).](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-35-320.jpg)
![Background Linear Algebra 17
1.3 CHANGE OF BASIS AND SIMILARITY
Change of basis and similarity are really about reformulating a given linear
algebra problem into an equivalent one that is easier to tackle. (It is a bit
like using equivalent frames of reference in the theory of relativity.) These
fundamental processes are reversible, so if we are able to answer the simpler
question, we can return with a solution to the initial problem.
Fix an n-dimensional vector space V and an (ordered) basis B =
{v1, v2, . . . , vn} for V. The co-ordinate vector of v ∈ V relative to B is
[v]B =
⎡
⎢
⎢
⎢
⎣
a1
a2
.
.
.
an
⎤
⎥
⎥
⎥
⎦
,
where the ai are the unique scalars for which v = a1v1 + a2v2 + · · · + anvn. If B
is another basis, we let [B , B] denote the change of basis matrix, that is the
n × n matrix whose columns are the co-ordinate vectors of the B basis vectors
relative to B. This is an invertible matrix with [B , B]−1
= [B, B ]. Co-ordinate
vectors now change according to the rule7
[v]B = [B, B ][v]B.
Now let T : V → V be a linear transformation. Its matrix [T]B relative
to the (ordered) basis B is defined as the n × n matrix whose columns are
the co-ordinate vectors [T(v1)]B, [T(v2)]B, . . . , [T(vn)]B of the images of
the B basis vectors. The reason why we work with columns8
rather than rows
here is that our transformations act on the left of vectors, and our composition
of two transformations is in accordance with this: (ST)(v) = S(T(v)). The
correspondence v → [v]B is a vector space isomorphism from V to n-space
Fn
. What T is doing, under this identification, is simply left multiplying column
vectors by the matrix [T]B:
[T(v)]B = [T]B[v]B.
(So, abstractly, a linear transformation is just left multiplication of column
vectors by a matrix.)
7. A good way to remember this and other change of basis results is that primed and unprimed
basis labels alternate.
8. This is the sensible rule, but unfortunately not all authors observe it. To break it invites trouble
in the more general setting of vector spaces over division rings. (In that setting, one should also
place the scalars on the right of vectors.)](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-44-320.jpg)
![18 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A
A permutation matrix is a square n × n matrix P whose rows (resp.
columns) are a permutation of the rows (resp. columns) of the identity
matrix I under some permutation p ∈ Sn (resp. p−1
∈ Sn). (Here, Sn is
the symmetric group of all permutations of 1, 2, . . . , n.) In terms of the
matrix of a linear transformation, and in the case of a row permutation
p, we have P = [T]B where B = {v1, v2, . . . , vn} is the standard basis of
Fn
and T : Fn
→ Fn
is the linear transformation whose action on B is
T(vi) = vp(i).
For fixed V and basis B, the correspondence T → [T]B provides
the fundamental isomorphism between the algebra L(V) of all linear
transformations of V (to itself) and the algebra Mn(F) of all n × n matrices
over F: it is a 1-1 correspondence that preserves sums, products9
and scalar
multiples. The result should be etched in the mind of every serious student of
linear algebra.10
Two square n × n matrices A and B are called similar if B = C−1
AC for some
invertible matrix C. “Similar” is an understatement here, because A and B will
haveidenticalalgebraicproperties. (Inparticular,similarmatriceshavethesame
eigenvalues, determinant, rank, trace,11
and so on.) This is because for a fixed
invertible C, and a variable matrix A, the conjugation mapping A → C−1
AC
is an algebra automorphism of Mn(F) (a 1-1 correspondence preserving
sums, products, and scalar multiples).12
And under an automorphism (or
isomorphism), an element and its image have the same algebraic properties.13
This view of similarity is entirely analogous to, for example, conjugation in
grouptheory.Butwhatisnewinthelinearalgebrasettingishownicelysimilarity
relatestothematricesofalineartransformationT : V → V ofann-dimensional
9. If we had put the co-ordinate vectors [T(vi)]B as rows of the representing matrix, the
correspondence would reverse products.
10. Unfortunately, nowadays some otherwise very good students come away from linear algebra
courses without ever having seen this.
11. The trace, tr A, of a square matrix A is the sum of its diagonal entries.
12. The so-called Skolem–Noether theorem of ring theory tells us that these conjugations are
the only algebra automorphisms of Mn(F). (See Jacobson’s Basic Algebra II, p. 222.)
13. Thinking of complex conjugation as an automorphism of C, we see that a complex number
and its conjugate are algebraically indistinguishable. In particular, there is really no such thing
as “the” (natural) complex number i satisfying i2
= −1, short of arbitrarily nominating one
of the two roots (because the two solutions are conjugates). This is unlike the distinction
between, say, the two square roots of 2 in R. Here, one root is positive, hence expressible as
a square of a real number; the other is not. So the two can be distinguished by an algebraic
property.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-45-320.jpg)
![Background Linear Algebra 19
space under a change of basis from B to B . They are always similar:
[T]B = C−1
[T]B C
where C = [B , B].
Moreover, every pair of similar matrices can be viewed as the matrices of a single
transformation relative to suitable bases.
Ausefulobservationinthecasethat C isapermutationmatrix,corresponding
to some permutation p ∈ Sn, but this time via the action of p on the columns of I,
is that C−1
AC is the matrix obtained by first permuting the columns of A under p,
and then permuting the rows of the resulting matrix by the same permutation p.
For instance, if p = (1 2 3) is the cyclic permutation, then
C =
⎡
⎣
0 1 0
0 0 1
1 0 0
⎤
⎦ and C−1
⎡
⎣
a b c
d e f
g h i
⎤
⎦ C =
⎡
⎣
i g h
c a b
f d e
⎤
⎦ .
A standard way of utilizing the transformation view of similarity, but with a
matrix outcome in mind, is this: suppose we are presented with an n × n matrix
A over the field F and we are looking for a simpler matrix B (perhaps diagonal)
to which A is similar. Firstly, let V = Fn
, let B be the standard basis for V,
and let T : V → V be the linear transformation that left multiplies column
vectors by A:
T
⎛
⎜
⎜
⎜
⎝
⎡
⎢
⎢
⎢
⎣
a1
a2
.
.
.
an
⎤
⎥
⎥
⎥
⎦
⎞
⎟
⎟
⎟
⎠
= A
⎡
⎢
⎢
⎢
⎣
a1
a2
.
.
.
an
⎤
⎥
⎥
⎥
⎦
.
Note [T]B = A. Secondly, “using one’s wits” (depending on additional
information about A), find another basis B relative to which the matrix
B = [T]B looks nice. Thirdly, let C = [B , B] be the change of basis matrix.
Note that C has the B basis vectors as its columns and is invertible. Now we
have our similarity B = C−1
AC by the change of basis result for the matrices of
a transformation.
Again, suppose T : V → V is a linear transformation of an n-dimensional
space. A subspace U of V is said to be invariant under T if T(U) ⊆ U (T
maps vectors of U into U). Notice that a nonzero vector v ∈ V is an eigenvector
of T (for some eigenvalue) precisely when v is invariant under T. (This
provides a clear geometric picture of why a proper rotation of the real plane](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-46-320.jpg)
![20 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A
about the origin through less than 180 degrees can’t have any eigenvalues—no
lines through the origin are invariant under the rotation.) If we choose a basis
B1 for an invariant subspace U and extend it to a basis B of V, then the matrix
of T relative to B is block upper triangular in which the top left block is m × m,
where m = dim U, and the bottom right block is (n − m) × (n − m) :
[T]B =
P Q
0 R
,
where P is the matrix of T|U : U → U relative to B1. This observation can
often be used as an inductive tool. (It also allows a neat noninductive proof
of the Cayley–Hamilton theorem in terms of transformations, by fixing v ∈ V
and taking U to be the subspace spanned by all the Ti
(v). Then through a
natural choice for B1, the matrix P is a “companion matrix” whose characteristic
polynomial p(x) ∈ Mm(F) is easily calculated, and for which p(T)(v) = 0 is
easily verified. The reader is invited to complete the argument, or to curse
the authors for not doing so!) The kernel and image of a transformation
T are always subspaces invariant under T. We record the following simple
generalization.
Proposition 1.3.1
Suppose S and T are commuting linear transformations of a vector space V. Then the
kernel and image of S are subspaces which are invariant under T.
Proof
Let U = ker S. For u ∈ U, we have
S(T(u)) = (ST)(u)
= (TS)(u) (by commutativity)
= T(S(u))
= T(0)
= 0 ,
which shows T(u) ∈ U. Thus, U is invariant under T. Similarly, so is S(V).
A vector space V is a direct sum of subspaces U1, U2, . . . , Uk, written
V = U1 ⊕ U2 ⊕ · · · ⊕ Uk,
if every v ∈ V can be written uniquely as v = u1 + u2 + · · · + uk, where each
ui ∈ Ui. In this case, a union of linearly independent subsets from each of the Ui](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-47-320.jpg)
![Background Linear Algebra 21
remains linearly independent. Consequently, dim V = dim U1 + dim U2 +
· · · + dim Uk. As with (internal) direct sums or products of other algebraic
structures, one can verify that a sum U1 + U2 + · · · + Uk of subspaces is a direct
sum, meaning U1 + U2 + · · · + Uk = U1 ⊕ U2 ⊕ · · · ⊕ Uk, by repeated use of
the condition that for k = 2, directness means that U1 ∩ U2 = 0. In general, we
check the “triangular conditions”:
U1 ∩ U2 = 0 ,
(U1 + U2) ∩ U3 = 0 ,
(U1 + U2 + U3) ∩ U4 = 0 ,
.
.
.
(U1 + U2 + U3 + · · · + Uk−1) ∩ Uk = 0 .
An especially useful observation (when teamed with results for change of
basis matrices) is the following.
Proposition 1.3.2
Suppose T : V → V is a linear transformation and
V = U1 ⊕ U2 ⊕ · · · ⊕ Uk
is a direct sum decomposition of V into T-invariant subspaces U1, U2, . . . , Uk. Pick a
basis Bi for each Ui and let B = B1 ∪ B2 ∪ · · · ∪ Bk. Then relative to the basis B for
V, the matrix of T is the block diagonal matrix
[T]B =
⎡
⎢
⎢
⎢
⎢
⎣
A1
A2
...
Ak
⎤
⎥
⎥
⎥
⎥
⎦
,
where Ai is the matrix relative to Bi of the restriction of T to Ui.
Proof
There is nothing to this if (1) we have a clear mental picture of what the matrix
of a transformation relative to a specified basis looks like,14
and (2) appreciate
that the restriction of a linear transformation T to an invariant subspace U is a
14. If one is constantly referring back to the definition of the matrix of a transformation, and
consulting with the “subscript doctor,” this distraction may hamper progress in later chapters.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-48-320.jpg)
![22 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A
linear transformation of U as a vector space in its own right. For instance, suppose
dim U1 = 3 and dim U2 = 2 and we label the basis vectors by B1 = {v1, v2, v3}
and B2 = {v4, v5}. Since U1 is T-invariant, for i = 1, 2, 3, the T(vi) are linear
combinations of only v1, v2, v3, so in the matrix [T]B, the first three columns have
zeros past row three. Similarly, for i = 4, 5, the T(vi) are linear combinations of
only v4, v5, so in the matrix [T]B, columns four and five have no nonzero entries
outside of rows four and five. And so on.
1.4 DIAGONALIZATION
There isn’t a question that one can’t immediately answer about a diagonal
matrix
D = diag(d1, d2, . . . , dn) =
⎡
⎢
⎢
⎢
⎣
d1 0 · · · 0
0 d2 0
.
.
.
...
0 0 · · · dn
⎤
⎥
⎥
⎥
⎦
.
For instance, its kth power is diag(dk
1, dk
2, . . . , dk
n). So it is of interest to know
when a square n × n matrix A is similar to a diagonal matrix. (Then, for example,
its powers can also be computed.) Such a matrix A is called diagonalizable:
there exists an invertible matrix C such that C−1
AC is diagonal. Standard
texts include many interesting applications of diagonalizable matrices, from
Markov processes, to finding principal axes of quadratic forms, through to
solving systems of first order linear differential equations. Later, in Chapter 6,
we examine an “approximate” version of diagonalization, which has modern
relevance to phylogenetics and multivariate interpolation.
Conceptually, the key to understanding diagonalization is through linear
transformations T : V → V. The matrix of T relative to a basis B is diagonal
precisely when the basis vectors are eigenvectors for various eigenvalues. In that
case, the matrix is simply the diagonal matrix of the matching eigenvalues, in
the order the basis vectors happen to be presented. An individual eigenvalue
will appear on the diagonal according to its algebraic multiplicity. (This is just
Proposition 1.3.2 when all the Ui are one-dimensional.) Sensibly, one should
reorder the basis vectors to group together those sharing the same eigenvalue.
To connect all this with matrices, we just use change of basis results.
Presented with a small n × n matrix A, whose eigenvalues we know, we can
test if A is diagonalizable by checking if the geometric multiplicities of its various
eigenvalues sum to n. And an explicit C that diagonalizes A can also be found.
Here is an example to remind us of the process.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-49-320.jpg)
![Background Linear Algebra 23
Example 1.4.1
Suppose we wish to diagonalize the real matrix
A =
⎡
⎣
3 1 1
1 3 1
1 1 3
⎤
⎦ .
Using a first row cofactor expansion, we see that the characteristic polynomial p(x)
of A is
p(x) = det
⎡
⎣
x − 3 −1 −1
−1 x − 3 −1
−1 −1 x − 3
⎤
⎦
= (x − 3)[(x − 3)2
− 1] + 1(−x + 3 − 1) − 1(1 + x − 3)
= (x − 2)2
(x − 5).
Hence the eigenvalues of A are 2, 5 with respective algebraic multiplicities 2 and 1.
We need to check if these agree with the geometric multiplicities. We can compute
a basis for the eigenspace E(2) using elementary row operations:
2I − A =
⎡
⎣
−1 −1 −1
−1 −1 −1
−1 −1 −1
⎤
⎦ −→
⎡
⎣
1 1 1
0 0 0
0 0 0
⎤
⎦ .
The corresponding homogeneous system (2I − A)x = 0 has two free variables,
from which we can pick out the basis
B2 =
⎧
⎨
⎩
⎡
⎣
−1
1
0
⎤
⎦ ,
⎡
⎣
−1
0
1
⎤
⎦
⎫
⎬
⎭
for E(2). For E(5) we proceed similarly:
5I−A =
⎡
⎣
2 −1 −1
−1 2 −1
−1 −1 2
⎤
⎦ −→
⎡
⎣
1 1 −2
−1 2 −1
2 −1 −1
⎤
⎦
−→
⎡
⎣
1 1 −2
0 3 −3
0 −3 3
⎤
⎦ −→
⎡
⎣
1 1 −2
0 1 −1
0 0 0
⎤
⎦ −→
⎡
⎣
1 0 −1
0 1 −1
0 0 0
⎤
⎦.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-50-320.jpg)
![24 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A
This gives one free variable in the homogeneous system (5I − A)x = 0, from which
we get the basis
B5 =
⎧
⎨
⎩
⎡
⎣
1
1
1
⎤
⎦
⎫
⎬
⎭
for the eigenspace E(5). Thus, the geometric multiplicities of the eigenvalues
2 and 5 sum to n = 3, whence A is diagonalizable.
We can diagonalize A explicitly with an invertible matrix C as follows. Let B be
the standard basis for V = F3 and note A is the matrix of its left multiplication map
of V relative to B. Next, form the basis for V
B = B2 ∪ B5 =
⎧
⎨
⎩
⎡
⎣
−1
1
0
⎤
⎦ ,
⎡
⎣
−1
0
1
⎤
⎦ ,
⎡
⎣
1
1
1
⎤
⎦
⎫
⎬
⎭
of eigenvectors of A. Finally take these basis vectors as the columns of the matrix
C =
⎡
⎣
−1 −1 1
1 0 1
0 1 1
⎤
⎦ .
The outcome, by the change of basis result for a linear transformation (looking
at the left multiplication map by A relative to B and noting that C = [B , B]), is
C−1
AC =
⎡
⎣
2 0 0
0 2 0
0 0 5
⎤
⎦ ,
a diagonal matrix having the eigenvalues 2 and 5 on the diagonal and repeated
according to their algebraic multiplicities.
In the above example, the diagonalization works over any field F whose
characteristic is not 3. (When F has characteristic 3, the above A has 2 as
its only eigenvalue but this has geometric multiplicity only 2, less than 3.)
So, in general, diagonalization depends on the base field. For instance, real
symmetric and complex hermitian matrices are always diagonalizable (in fact by
an orthogonal and unitary matrix, respectively), but over the two element field,
only the idempotent matrices E (those satisfying E2
= E) are diagonalizable.
A frequently used observation is that an n × n matrix that has n distinct
eigenvalues is diagonalizable. The general theorem is that A ∈ Mn(F) is
diagonalizable if and only if the minimal polynomial of A factors into distinct
linear factors. The minimal polynomial is the unique monic polynomial m(x)](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-51-320.jpg)
![Background Linear Algebra 25
ofleastdegreesuchthatm(A) = 0.Itcanbecalculatedbyfindingthefirstpower
As
of A that is linearly dependent on the earlier powers I, A, A2
, . . . , As−1
, say
As
= c0I + c1A + · · · + cs−1As−1
, and taking
m(x) = xs
− cs−1xs−1
− · · · − c1x − c0.
The minimal polynomial divides all other polynomials that vanish at A. In
particular, by the Cayley–Hamilton theorem, the minimal polynomial divides
the characteristic polynomial, so the degree of m(x) is at most n. In fact, m(x)
has the same zeros as the characteristic polynomial (the eigenvalues of A),
only with smaller multiplicities. In some ways, the minimal polynomial is more
revealing of the properties of a matrix than the characteristic polynomial. One
can also show that, as an ideal of F[x], the kernel of the polynomial evaluation
map f → f (A) has the minimal polynomial of A as its monic generator. This is
as good a place as any to record another property of the minimal polynomial,
which we use (sometimes implicitly) in later chapters.
Proposition 1.4.2
The dimension of the subalgebra F[A] generated by a square matrix A ∈ Mn(F) agrees
with the degree of the minimal polynomial m(x) of A.
Proof
Finite-dimensionality of Mn(F) guarantees some power of A is dependent on earlier
powers, so there is a least such power As that is so dependent. Let
(∗) As
= c0I + c1A + · · · + cs−1As−1
be the corresponding dependence relation. Now I, A, A2, . . . , As−1 all lie in F[A]
and are linearly independent by choice of s. We need only show they span F[A]
in order to conclude they form a basis with s members, whence dim F[A] = s =
deg(m(x)). In turn, since the powers of A span F[A], it is enough to get these
powersaslinearcombinationsofI, A, A2, . . . , As−1.Butthisjustinvolvesrepeated
applications of the relationship (∗):
As+1
= AAs
= A(c0I + c1A + · · · + cs−1As−1
)
= c0A + c1A2
+ · · · + cs−1As
= c0A + c1A2
+ · · · + cs−2As−1
+ cs−1(c0I + c1A + · · · + cs−1As−1
)
= cs−1c0I + (c0 + cs−1c1)A + · · · + (cs−2 + c2
s−1)As−1
and so on.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-52-320.jpg)
![26 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A
Unlike the minimal polynomial of an algebraic field element, minimal
polynomials of matrices need not be irreducible. In fact, any monic polynomial
of positive degree can be the minimal polynomial of a suitable matrix, and
the same can happen for the characteristic polynomial.15
The following is the
standard example:
Example 1.4.3
Let f (x) = xn + cn−1xn−1 + · · · + c2x2 + c1x + c0 ∈ F[x] be a monic polynomial
of degree n. Then the following “companion matrix”
C =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 · · · −c0
1 0 −c1
0 1 −c2
.
.
.
...
.
.
.
1 0 −cn−2
0 0 · · · 0 1 −cn−1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
has f (x) as both its minimal and characteristic polynomials. To see this, one
observes that the first n powers of C are independent so the minimal polynomial
has degree n and necessarily agrees with the characteristic polynomial. On the other
hand, the characteristic polynomial det(xI − C) can easily be computed to be f (x)
directly, by a cofactor expansion in the first row (combined with induction for
matrices of size n − 1 when evaluating the (1, 1) cofactor).
Another useful point of view of diagonalizable matrices A ∈ Mn(F) is that
they are precisely the matrices possessing a “spectral resolution”:
A = λ1E1 + λ2E2 + · · · + λkEk
where the λi are scalars and the Ei are orthogonal idempotent matrices, that is,
E2
i = Ei and EiEj = 0 for i = j. (In the spectral resolution, there is no loss of
generality in assuming that the λi are distinct, in which case the Ei are actually
polynomials in A.) This is a nice “basis-free” approach.
15. Consequently, there can be no way of exactly computing the eigenvalues of a general matrix.
Nor can there be a “formula” for the eigenvalues in terms of the rational operations of addition,
multiplication, division, and extraction of mth roots on the entries of a general matrix of size
bigger than 4 × 4. (This follows from Galois theory, more particularly the Abel–Ruffini theorem
that quintic and higher degree polynomial equations are not “solvable by radicals.”) However,
the matrices that arise in practice (e.g., tridiagonal) are often amenable to fast, high-precision,
eigenvalue methods.](https://image.slidesharecdn.com/1190618-250523170940-b0c50984/85/Advanced-Topics-In-Linear-Algebra-Weaving-Matrix-Problems-Through-The-Weyr-Form-Kevin-Omeara-53-320.jpg)
Advanced Topics In Linear Algebra Weaving Matrix Problems Through The Weyr Form Kevin Omeara
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- 6. Advanced Topics in Linear Algebra
- 7. This page intentionally left blank
- 8. Advanced Topics in Linear Algebra Weaving Matrix Problems through the Weyr Form KEVIN C. O’MEARA JOHN CLARK CHARLES I. VINSONHALER 3
- 9. 3 Oxford University Press, Inc., publishes works that further Oxford University’s objective of excellence in research, scholarship, and education. Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Copyright © 2011 by Oxford University Press Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 www.oup.com Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data O’Meara, Kevin C. Advanced topics in linear algebra : weaving matrix problems through the Weyr Form / Kevin C. O’Meara, John Clark, Charles I. Vinsonhaler. p. cm. Includes bibliographical references and index. ISBN 978-0-19-979373-0 1. Algebras, Linear. I. Clark, John. II. Vinsonhaler, Charles Irvin, 1942- III. Title. QA184.2.O44 2011 512’.5-dc22 2011003565 9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper
- 10. DEDICATED TO Sascha, Daniel, and Nathania Kevin O’Meara Austina and Emily Grace Clark John Clark Dorothy Snyder Vinsonhaler Chuck Vinsonhaler
- 11. This page intentionally left blank
- 12. CONTENTS Preface xi Our Style xvii Acknowledgments xxi PARTONE:The Weyr Form and Its Properties 1 1. Background Linear Algebra 3 1.1. The Most Basic Notions 4 1.2. Blocked Matrices 11 1.3. Change of Basis and Similarity 17 1.4. Diagonalization 22 1.5. The Generalized Eigenspace Decomposition 27 1.6. Sylvester’s Theorem on the Matrix Equation AX − XB = C 33 1.7. Canonical Forms for Matrices 35 Biographical Notes on Jordan and Sylvester 42 2. The Weyr Form 44 2.1. What Is the Weyr Form? 46 2.2. Every Square Matrix Is Similar to a Unique Weyr Matrix 56 2.3. Simultaneous Triangularization 65 2.4. The Duality between the Jordan and Weyr Forms 74 2.5. Computing the Weyr Form 82 Biographical Note on Weyr 94 3. Centralizers 96 3.1. The Centralizer of a Jordan Matrix 97 3.2. The Centralizer of a Weyr Matrix 100 3.3. A Matrix Structure Insight into a Number-Theoretic Identity 105 3.4. Leading Edge Subspaces of a Subalgebra 108
- 13. viii Contents 3.5. Computing the Dimension of a Commutative Subalgebra 114 Biographical Note on Frobenius 123 4. The Module Setting 124 4.1. A Modicum of Modules 126 4.2. Direct Sum Decompositions 135 4.3. Free and Projective Modules 144 4.4. Von Neumann Regularity 152 4.5. Computing Quasi-Inverses 159 4.6. The Jordan Form Derived Module-Theoretically 169 4.7. The Weyr Form of a Nilpotent Endomorphism: Philosophy 174 4.8. The Weyr Form of a Nilpotent Endomorphism: Existence 178 4.9. A Smaller Universe for the Jordan Form? 185 4.10. Nilpotent Elements with Regular Powers 188 4.11. A Regular Nilpotent Element with a Bad Power 195 Biographical Note on Von Neumann 197 PARTTWO:Applications of the Weyr Form 199 5. Gerstenhaber’s Theorem 201 5.1. k-Generated Subalgebras and Nilpotent Reduction 203 5.2. The Generalized Cayley–Hamilton Equation 210 5.3. Proof of Gerstenhaber’s Theorem 216 5.4. Maximal Commutative Subalgebras 221 5.5. Pullbacks and 3-Generated Commutative Subalgebras 226 Biographical Notes on Cayley and Hamilton 236 6. Approximate Simultaneous Diagonalization 238 6.1. The Phylogenetic Connection 241 6.2. Basic Results on ASD Matrices 249 6.3. The Subalgebra Generated by ASD Matrices 255 6.4. Reduction to the Nilpotent Case 258 6.5. Splittings Induced by Epsilon Perturbations 260 6.6. The Centralizer of ASD Matrices 265 6.7. A Nice 2-Correctable Perturbation 268 6.8. The Motzkin–Taussky Theorem 271 6.9. Commuting Triples Involving a 2-Regular Matrix 276 6.10. The 2-Regular Nonhomogeneous Case 287 6.11. Bounds on dim C[A1, . . . , Ak] 297 6.12. ASD for Commuting Triples of Low Order Matrices 301 Biographical Notes on Motzkin and Taussky 307
- 14. Contents ix 7. Algebraic Varieties 309 7.1. Affine Varieties and Polynomial Maps 311 7.2. The Zariski Topology on Affine n-Space 320 7.3. The Three Theorems Underpinning Basic Algebraic Geometry 326 7.4. Irreducible Varieties 328 7.5. Equivalence of ASD for Matrices and Irreducibility of C(k, n) 339 7.6. Gerstenhaber Revisited 342 7.7. Co-Ordinate Rings of Varieties 347 7.8. Dimension of a Variety 353 7.9. Guralnick’s Theorem for C(3, n) 364 7.10. Commuting Triples of Nilpotent Matrices 370 7.11. Proof of the Denseness Theorem 378 Biographical Notes on Hilbert and Noether 381 Bibliography 384 Index 390
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- 16. PREFACE “Oldhabitsdiehard.”ThismaximmayhelpexplainwhytheWeyrformhasbeen almost completely overshadowed by its cousin, the Jordan canonical form. Most have never even heard of the Weyr form, a matrix canonical form discovered by the Czech mathematician Eduard Weyr in 1885. In the 2007 edition of the Handbook of Linear Algebra, a 1,400-page, authoritative reference on linear algebra matters, there is not a single mention of the Weyr form (or its associated Weyr characteristic). But this canonical form is as useful as the Jordan form, which was discovered by the French mathematician Camille Jordan in 1870. Our book is in part an attempt to remedy this unfortunate situation of a grossly underutilized mathematical tool, by making the Weyr form more accessible to those who use linear algebra at its higher level. Of course, that class includes most mathematicians, and many others as well in the sciences, biosciences, and engineering. And we hope our book also helps popularize the Weyr form by demonstrating its topical relevance, to both “pure” and “applied” mathematics. We believe the applications to be interesting and surprising. Althoughtheunifyingthemeofourbookisthedevelopmentandapplications of the Weyr form, this does not adequately describe the full scope of the book. The three principal applications—to matrix commutativity problems, approximate simultaneous diagonalization, and algebraic geometry—bring the reader right up to current research (as of 2010) with a number of open questions, and also use techniques and results in linear algebra not involving canonicalforms.Andevenintopicsthatarefamiliar,wepresentsomeunfamiliar results, such as improving on the known fact that commuting matrices over an algebraically closed field can be simultaneously triangularized. Matrix canonical forms (with respect to similarity) provide exemplars for each similarity class of square n × n matrices over a fixed field. Their aesthetic qualities have long been admired. But canonical forms also have some very concrete applications. The authors were drawn to the Weyr form through a
- 17. xii Preface question that arose in phylogenetic invariants in biomathematics in 2003. Prior to that, we too were completely unaware of the Weyr form. It has been a lot of fun rediscovering the lovely properties of the Weyr form and, in some instances, finding new properties. In fact, quite a number of results in our book have (apparently) not appeared in the literature before. There is a wonderful mix of ideas involved in the description, derivation, and applications of the Weyr form: linear algebra, of course, but also commutative and noncommutative ring theory, module theory, field theory, topology (Euclidean and Zariski), and algebraic geometry. We have attempted to blend these ideas together throughout our narrative. As much as possible, given the limits of space, we have given self-contained accounts of the nontrivial results we use from outside the area of linear algebra, thereby making our book accessible to a good graduate student. For instance, we develop from scratch a fair bit of basic algebraic geometry, which is unusual in a linear algebra book. If nothing else, we claim to have written quite a novel linear algebra text. We are not aware of any book with a significant overlap with the topics in ours, or of any book that devotes an entire chapter to the Weyr form. However, Roger Horn recently informed us (in September 2009) that the upcoming second edition of the Horn and Johnson text Matrix Analysis will have a section on the Weyr form in Chapter 3. Of course, whether our choice of topics is good or bad, and what sort of job we have done, must ultimately be decided by the reader. All seven chapters of our book begin with a generous introduction, as do most sections within a chapter. We feel, therefore, that there is not a lot of point in describing the chapter contents within this preface, beyond the barest summary that follows. Besides, the reader is not expected to know what the Weyr form is at this time. PART I: THE WEYR FORM AND ITS PROPERTIES 1: Background Linear Algebra We do a quick run-through of some of the more important basic concepts we require from linear algebra, including diagonalization of matrices, the description of the Jordan form, and desirable features of canonical matrix forms in general. 2: The Weyr Form Here we derive the Weyr form from scratch, establish its basic properties, and detail an algorithm for computing the Weyr form of nilpotent matrices (always the core case). We also derive an important duality between the Jordan and Weyr structures of nilpotent matrices. 3: Centralizers The matrices that centralize (commute with) a given nilpotent Jordan matrix have a known explicit description. Here we do likewise for the
- 18. Preface xiii Weyr form, for which the centralizer description is simpler. It is this property that gives the Weyr form an edge over its Jordan counterpart in a number of applications. 4: The Module Setting The Jordan form has a known ring-theoretic derivation through decompositions of finitely generated modules over principal ideal domains. In this chapter we derive the Weyr form ring theoretically, but in an entirely different way, by using ideas from decompositions of projective modules over von Neumann regular rings. The results suggest that the Weyr form lives in a somewhat bigger universe than its Jordan counterpart, and is perhaps more natural. PART II: APPLICATIONS OF THE WEYR FORM 5: Gerstenhaber’s Theorem The theorem states that the subalgebra F[A, B] generated by two commuting n × n matrices A and B over a field F has dimension at most n. It was first proved using algebraic geometry, but later Barría and Halmos, and Laffey and Lazarus, gave proofs using only linear algebra and the Jordan form. Here we simplify the Barría–Halmos proof even further through the use of the Weyr form in tandem with the Jordan form, utilizing an earlier duality. 6: Approximate Simultaneous Diagonalization Complex n × n matrices A1, A2, . . . , Ak are called approximately simultaneously diagonalizable (ASD) if they can be perturbed to simultaneously diagonalizable matrices B1, B2, . . . , Bk. In this chapter we attempt to show how the Weyr form is a promising tool (more so than the Jordan form) for establishing ASD of various classes of commuting matrices using explicit perturbations. The ASD property has been used in the study of phylogenetic invariants in biomathematics, and multivariate interpolation. 7: Algebraic Varieties Here we give a largely self-contained account of the algebraic geometry connection to the linear algebra problems studied in Chapters 5 and 6. In particular, we cover most of the known results on the irreducibility of the variety C(k, n) of all k-tuples of commuting complex n × n matrices. The Weyr form is used to simplify some earlier arguments. Irreducibility of C(k, n) is surprisingly equivalent to all k commuting complex n × n matrices having the ASD property described in Chapter 6. But a number of ASD results are known only through algebraic geometry. Some of this work is quite recent (2010).
- 19. xiv Preface Our choice of the title Advanced Topics in Linear Algebra indicates that we are assuming our reader has a solid background in undergraduate linear algebra (see the introduction to Chapter 1 for details on this). However, it is probably fair to say that our treatment is at the higher end of “advanced” but without being comprehensive, compared say with Roman’s excellent text Advanced Linear Algebra,1 in the number of topics covered. For instance, while some books on advanced linear algebra might take the development of the Jordan form as one of their goals, we assume our readers have already encountered the Jordan form (although we remind readers of its properties in Chapter 1). On the other hand, we do not assume our reader is a specialist in linear algebra. The book is designed to be read in its entirety if one wishes (there is a continuous thread), but equally, after a reader has assimilated Chapters 2 and 3, each of the four chapters that follow Chapter 3 can be read in isolation, depending on one’s “pure” or “applied” leanings. At the end of each chapter, we give brief biographical sketches of one or two of the principal architects of our subject. It is easy to forget that mathematics has been, and continues to be, developed by real people, each generation building on the work of the previous—not tearing it down to start again, as happens in many other disciplines. These sketches have been compiled from various sources, but in particular from the MacTutor History of Mathematics web site of the University of St. Andrews, Scotland [http://www-history.mcs.st- andrews.ac.uk/Biographies],andI.Kleiner’sAHistoryofAbstractAlgebra.Note, however, that we have given biographies only for mathematicians who are no longer alive. When we set out to write this book, we were not thinking of it as a text for a course, but rather as a reference source for teachers and researchers. But the more we got into the project, the more apparent it became that parts of the book would be suitable for graduate mathematics courses (or fourth-year honorsundergraduatecourses,inthecaseofthebetterantipodeanuniversities). True, we have not included exercises (apart from a handful of test questions), but the nature of the material is such that an instructor would find it rather easy (and even fun) to make up a wide range of exercises to suit a tailored course. As to the types of course, a number spring to mind: (1) A second-semester course following on from a first-year graduate course in linear algebra, covering parts of Chapters 1, 2, 3, and 6. 1. Apart from our background in Chapter 1, there is no overlap in the topics covered in our book and that of Roman.
- 20. Preface xv (2) A second-semester course following on from an abstract algebra course that covered commutative and noncommutative rings, covering parts of Chapters 1, 2, 3, 4, 5, and 7. (3) The use of Chapter 4 as a supplement in a course on module theory. (4) The use of Chapter 7 as a supplement in a course on algebraic geometry or biomathematics (e.g., phylogenetics). The authors welcome comments and queries from readers. Please use the following e-mail addresses: staf198@ext.canterbury.ac.nz (Kevin O’Meara) jclark@maths.otago.ac.nz (John Clark) charles.vinsonhaler@uconn.edu (Chuck Vinsonhaler)
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- 22. OUR STYLE Mathematicians are expected to be very formal in their writings. An unintended consequence of this is that mathematics has more than its share of rather boring, pedantic, and encyclopedic books—good reading for insomniacs. We have made a conscious decision to write in a somewhat lighter and more informal style. We comment here on some aspects of this style, so that readers will know what to expect. The mathematical content of our arguments, on the other hand, is always serious. Some mathematics writers believe that because they have formally spelled out all the precise definitions of every concept, often lumped together at the very beginning of a chapter or section, the reader must be able to understand and appreciate their arguments. This is not our experience. (One first expects to see the menu at a restaurant, not a display of all the raw ingredients.) And surely, if a result is stated in its most general form, won’t the reader get an even bigger insight into the wonders of the concepts? Mistaken again, in our view, because this may obscure the essence of the result. To complete the trifecta of poor writing, mathematicians sometimes try to tell the reader everything they know about a particular topic; in so doing, they often cloud perspective. We have kept the formal (displayed) statements of definitions to a minimum— reserved for the most important concepts. We have also attempted to delay the formal definition until after suitable motivation of the concept. The concept is usually then illustrated by numerous examples. And in the development proper, we don’t tell everything we know. In fact, we often invite (even challenge) the reader to continue the exploration, sometimes in a footnote.
- 23. xviii Our Style We make no apology for the use of whimsy.1 In our view, there is a place for whimsy even within the erudite writings of mathematicians. It can help put a human face on the authors (we are not high priests) and can energize a reader to continue during the steeper climbs. Our whimsical comments are mostly reserved for an occasional footnote. But footnotes, being footnotes, can be skipped without loss of continuity to the story. We have tried to avoid the formality of article writing in referencing works. Thus, rather than say “ see Proposition 4.8 (2) and the Corollary on p. 222 of [BAII] ” we would tend to say simply “ see Chapter 4 of Jacobson’s Basic Algebra II.” Likewise, an individual paper by Joe Blog that is listed in our bibliography will usually be referred to as “the 2003 paper by Blog,” if there is only one such paper. The interested reader can then consult the source for more detail. What constitutes “correct grammar” has been a source of much discussion and ribbing among the three authors, prompted by their different education backgrounds (New Zealand, Scotland, and U.S.A.). By and large, the British Commonwealth has won out in the debate. But we are conscious, for example, of the difference in the British and American use of “that versus which,”2 and in punctuation. So please bear with us. Our notation and terminology are fairly standard. In particular, we don’t put brackets around the argument in the dimension dim V of a vector space V or the rank of a matrix A, rank A. However, we do use brackets if both the mathematical operator and argument are in the same case. Thus, we write ker(b) and im(p) for the kernel and image of module homomorphisms b and p, rather than the visually off-putting ker b and im p. Undoubtedly, there will be some inconsistencies in our implementation of this policy. An index entry such as Joe Blog’s theorem, 247, 256, 281 indicates that the principal statement or discussion of the theorem can be found on page 256, the one in boldface. This is done sparingly, reserved for the most important concepts, definitions, or results. Very occasionally, an entry may have more than one boldfaced page to indicate the most important, but separate, treatments of a topic. Finally, a word to a reader who perceives some “cheerleading” on our part when discussing the Weyr form. We have attempted to be even-handed in our 1. In a 2009 interview (by Geraldine Doogue, Australian ABC radio), Michael Palin (of Monty Python and travel documentary fame, and widely acclaimed as a master of whimsy) was asked why the British use whimsy much more so than Americans. His reply, in part, was that Britain has had a more settled recent history. America has been more troubled by wars and civil rights. Against this backdrop, Americans have tended to take things more seriously than the British. 2. Our rule is “that” introduces a defining clause, whereas “which” introduces a nondefining clause.
- 24. Our Style xix treatment of the Weyr and Jordan forms (the reader should find ample evidence of this). But when we are very enthused about a particular result or concept, we tell our readers so. Wouldn’t life be dull without such displays of feeling? Unfortunately, mathematics writers often put a premium on presenting material in a deadpan, minimalist fashion.
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- 26. ACKNOWLEDGMENTS These fall into two groups : (1) A general acknowledgment of those people who contributedtothemathematicsofourbookoritspublishing,and(2)Apersonal acknowledgment from each of the three authors of those who have given moral and financial support during the writing of the book, as well as a recognition of those who helped support and shape them as professional mathematicians over some collective 110 years! We are most grateful to Mike Steel (University of Canterbury, New Zealand) for getting us interested in the linear algebra side of phylogenetics, and to Elizabeth Allman (University of Alaska, Fairbanks) for contributing the section on phylogenetics in Chapter 6. Many thanks to Paul Smith (University of Washington, Seattle) for supplying the proof of a Denseness Theorem in Chapter 7. We also single out Roger Horn (University of Utah, Salt Lake City) for special thanks. His many forthright, informative comments on an earlier draft of our book, and his subsequent e-mails, have greatly improved the final product. It is a pleasure to acknowledge the many helpful comments, reference sources, technical advice, and the like from other folk, particularly Pere Ara, John Burke, Austina Clark, Herb Clemens, Keith Conrad, Ken Goodearl, Robert Guralnick, John Hannah, John Holbrook, Robert Kruse, James Milne, Ross Moore, Miki Neumann, Keith Nicholson, Vadim Olshevsky, Matja Omladič, Bob Plemmons, John Shanks, Boris Shekhtman, Klemen Šivic, Molly Thomson, Daniel Velleman (Editor of American Mathematical Monthly), Graham Wood, and Milen Yakimov. To the four reviewers who reported to Oxford University Press on an earlier draft of our book, and who made considered, insightful comments, we say thank you very much. In particular, we thank the two of you who suggested that our original title The Weyr Form: A Useful Alternative to the Jordan Canonical Form did not convey the full scope of our book.
- 27. xxii Acknowledgments Finally, our sincere thanks to editor Phyllis Cohen and her assistant Hallie Stebbins, production editor Jennifer Kowing, project manager Viswanath Prasanna from Glyph International, and the rest of the Oxford University Press (New York) production team (especially the copyeditor and typesetter) for their splendid work and helpful suggestions. They freed us up to concentrate on the writing, unburdened by technical issues. All queries from us three greenhorns were happily and promptly answered. It has been a pleasure working with you. From Kevin O’Meara. The biggest thanks goes to my family, of whom I am so proud: wife Leelalai, daughters Sascha and Nathania, and son Daniel. They happily adopted a new member into the family, “the book.” Thanks also to those who fed and sheltered me during frequent trips across the Tasman (from Brisbane to Christchurch and Dunedin), and across the Pacific (from Christchurch to Storrs, Connecticut), in connection with the book (or its foundations): John and Anna-Maree Burke, Brian and Lynette O’Meara, Lloyd andPatriciaAshby, ChuckandPattyVinsonhaler, MikeandSusanStuart, John and Austina Clark, Gabrielle and Murray Gormack. I have had great support from the University of Connecticut (U.S.A.) during my many visits over the last 30 years, particularly from Chuck Vinsonhaler and Miki Neumann. The University of Otago, New Zealand (host John Clark) has generously supported me during the writing of this book. Many fine mathematicians have influenced me over the years: Pere Ara, Richard Brauer, Ken Goodearl, Israel Herstein, Nathan Jacobson, Robert Kruse (my Ph.D. adviser), and James Milne, to name a few. I have also received generous support from many mathematics secretaries and technical staff, particularly in the days before I got round to learning L A TEX : Ann Tindal, Tammy Prentice, Molly Thomson, and John Spain are just four representative examples. Finally, I thank Gus Oliver for his unstinting service in restoring the health of my computer after its bouts of swine ’flu. From John Clark. First and foremost, I’m most grateful to Kevin and Chuck for inviting me on board the good ship Weyr form. Thanks also to the O’Meara family for their hospitality in Brisbane and to the Department of Mathematics and Statistics of the University of Otago for their financial support. Last, but certainly not least, my love and gratitude to my wife Austina for seeing me through another book. From Chuck Vinsonhaler. I am grateful to my wife Patricia for her support, and thankful for the mathematical and expository talents of my coauthors.
- 28. PART ONE The Weyr Form and Its Properties In the four chapters that compose the first half of our book, we develop the Weyr form and its properties, starting from scratch. Chapters 2 and 3 form the core of this work. Chapter 1 can be skipped by readers with a solid background in linear algebra, while Chapter 4, which gives a ring-theoretic derivation of the Weyr form, is optional (but recommended) reading. Applications involving the Weyr form come in Part II.
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- 30. 1 Background Linear Algebra Most books in mathematics have a background starting point. Ours is that the reader has had a solid undergraduate (or graduate) education in linear algebra. In particular, the reader should feel comfortable with an abstract vector space over a general field, bases, dimension, matrices, determinants, linear transformations, change of basis results, similarity, eigenvalues, eigenvectors, characteristic polynomial, the Cayley–Hamilton theorem, direct sums, diagonalization, and has at least heard of the Jordan canonical form. What is important is not so much a knowledge of results in linear algebra as an understanding of the fundamental concepts. In actual fact, there are rather few specific results needed as a prerequisite for understanding this book. The way linear algebra is taught has changed greatly over the last 50 years, and mostly for the better. Also, what was once taught to undergraduates is now often taught to graduate students. Prior to around the 1990s, linear algebra was at times presented as the poor cousin to calculus (or analysis). In some circles that view persists, but now most agree that linear algebra rivals calculus in applicability. (Every time one “Googles,” there is a calculation of a principal eigenvector of a gigantic matrix, of order several billion, to determine
- 31. 4 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A page rank. Amazing!1 ) This book was also motivated by an application, to phylogenetics, as discussed in Chapter 6. The authors are of the school that believes not teaching linear transforma- tions in linear algebra courses is to tell only half the story, even if one’s primary applications are to matrices. The full power of a linear algebra argument often comes from flipping back and forth between a matrix view and a transformation view. Without linear transformations, many similarity results for matrices lose their full impact. Also, the concept of an invariant subspace of a linear transformation is one of the most central in all linear algebra. (As a special case, the notion of an eigenvector corresponds to a 1-dimensional invariant subspace.) In this chapter, we will quickly run through a few of the more important basic concepts we require, but not in any great depth, with very few proofs, and sometimes scant motivation. The concepts are covered in many, many texts. The reader who wants more detail may wish to consult his or her own favorites. Ours include the books by Kenneth Hoffman and Ray Kunze (Linear Algebra), Roger Horn and Charles Johnson (Matrix Analysis), and Keith Nicholson (Linear Algebra with Applications). But there are many other fine books on linear algebra. Our advice to a reader who is already comfortable with the basics of linear algebra (as outlined in the opening paragraph) is to proceed directly to Chapter 2, and return only to check on notation, etc., should the need arise. 1.1 THE MOST BASIC NOTIONS It’s time to get down to the nitty-gritty, beginning with a summary of basic notions in linear algebra. In the first few pages, it is hard to avoid the unexciting format of recalling definitions, registering notation, and blandly stating results. In short, the things mathematics books are renowned for? Bear with us—our treatment will lighten up later in the chapter, when we not only recall concepts but also (hopefully) convey our particular slant on them. In basic calculus and analysis, there is probably not a great variation in how two (competent) individuals view the material. The mental pictures are pretty much the same. But it is less clear what is going on inside a linear algebraist’s head. The authors would venture to say that there is more variation in how individuals view the subject matter of linear algebra. (For instance, some get by in linear algebra without using linear transformations, although it would be extremely rare for someone in calculus to never use functions.) It often depends on an individual’s particular background in other mathematics. 1. See, for example, the 2006 article “The $25,000,000,000 eigenvector: The linear algebra behind Google,” by K. Bryan and T. Liese.
- 32. Background Linear Algebra 5 Thatisnottosayoneviewoflinearalgebraisrightandanotherisflatlywrong. If there are particular points of view that come across in the present book, their origins probably lie in the time the authors have worked with algebraic structures (such as semigroups, groups, rings, and associative algebras), and with having developed a healthy respect for such disciplines as category theory, universal algebra, and algebraic geometry. Of course, we have also come to admire the beautiful concepts in analysis and topology, some of which we use in Chapters 6 and 7. In this respect, our philosophical view is mainstream— mathematics, of all disciplines, should never be compartmentalized. The letter F will denote a field, usually algebraically closed (such as the complex field C), that is, every polynomial over F of positive degree has roots in F. The space of all n-tuples (which we usually write as column vectors) of elements from F is denoted by Fn . This space is the model for all n-dimensional vector spaces over F, because every n-dimensional space is isomorphic to Fn . By the standard basis for Fn we mean the basis ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 0 . . . 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 . . . 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 1 . . . 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , · · · , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 . . . 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ . The ring of polynomials, in the indeterminate x and with coefficients from F, is denoted by F[x]. This ring plays a similar role in linear algebra to that of the ring Z of integers in group theory. (Both rings are Euclidean domains and, for instance, the order of a group element, as an element of the ring Z, translates to the minimal polynomial of a matrix, as an element of the ring F[x].) If V is a vector space over F (usually finite-dimensional), its dimension is denoted by dim V. The subspace of V spanned (or generated) by vectors v1, v2, . . . , vn is denoted by v1, v2, . . . , vn. If U1, U2, . . . , Uk are subspaces of V, their sum is the subspace U1 + U2 + · · · + Uk = {v ∈ V : v = u1 + u2 + · · · + uk for some ui ∈ Ui}. For a linear transformation T : V → W from one vector space V to another W, the rank of T is the dimension of the image T(V), and the nullity of T is the dimension of the null space or kernel, ker T = {v ∈ V : T(v) = 0}. We have the fundamental rank, nullity connection: rank T + nullity T = dim V.
- 33. 6 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A For an m × n matrix A, this translates as rank A + nullity A = n = the number of columns of A, where nullity A is the dimension of the solution space of the homogeneous system Ax = 0, and rank A is either column rank or row rank (maximum number of independent columns or rows; they are the same). The matrix A is said to have full column-rank if rank A = n (its columns are linearly independent). The set of all n × n (square) matrices over F is denoted by Mn(F). The arithmetic of Mn(F) under addition, multiplication, and scalar multiplication is the most natural model of noncommutative (but associative) arithmetic.2 This is one of the principal reasons why linear algebra is such a powerful tool.3 So much of linear algebra and its applications revolve around the concepts of eigenvalues and eigenvectors. Our book is no exception. An eigenvalue of a matrix A ∈ Mn(F), or a linear transformation T : V → V, is a scalar λ ∈ F such that Av = λv or T(v) = λv for some nonzero vector v (in Fn or V, respectively). Any such v is called an eigenvector of A or T corresponding to the eigenvalue λ. The eigenspace of A corresponding to λ is E(λ) = ker(λI − A), which is just the set of all eigenvectors corresponding to λ together with the zero vector. (Here I is the identity matrix in Mn(F).) By the geometric multiplicity of λ we mean the dimension of E(λ). The characteristic polynomial of A is p(x) = det(xI − A). Although this polynomial far from “characterizes” the matrix A, it does reflect many of its important properties. For instance, the zeros of p(x) are precisely the eigenvalues of A. The reason why we often restrict F to being algebraically closed is to ensure eigenvalues always exist. In this case, if λ1, λ2, . . . , λk are the distinct eigenvalues of A and p(x) = (x − λ1)m1 (x − λ2)m2 · · · (x − λk)mk is the factorization of the characteristic polynomial into linear factors, then mi is called the algebraicmultiplicity of the eigenvalue λi (and m1 + m2 + · · · + mk = n). The geometric multiplicity of an eigenvalue can never exceed its algebraic multiplicity. A frequently used observation is that the eigenvalues of a triangular matrix are its diagonal entries. 2. The so-called Wedderburn–Artin theorem of ring theory more or less confirms this if F is algebraically closed: the Mn(F) are the only simple, finite-dimensional associative algebras, and finite direct products of these algebras give all “well-behaved” finite-dimensional algebras. 3. Another reason for the success of linear algebra, of course, is that it is suited to studying linear approximation problems.
- 34. Background Linear Algebra 7 For a polynomial f (x) = a0 + a1x + · · · + amxm ∈ F[x] and square matrix A ∈ Mn(F), we can form the matrix polynomial f (A) = a0I + a1A + a2A2 + · · · + amAm . This polynomial evaluation map f → f (A) (for a fixed A), from F[x] to Mn(F), is a simple but useful algebra homomorphism (i.e., a linear mapping that preserves multiplication). The Cayley–Hamilton theorem says that over any field F, every square matrix A vanishes at its characteristic polynomial: p(A) = 0 where p is the characteristic polynomial of A. The square matrices A that have 0 as an eigenvalue are the singular (noninvertible) matrices, because they are the matrices for which the homogeneous system Ax = 0 has a nonzero solution. A square matrix A is called nilpotent if Ar = 0 for some r ∈ N and in this case the least such r is the (nilpotency) index of A. If A is nilpotent then 0 is its only eigenvalue. (For if Ar = 0 and Ax = λx for some nonzero x, then 0 = Ar x = λr x, which implies λ = 0.) Over an algebraically closed field, the converse also holds, as we show in our first proposition: Proposition 1.1.1 Over an algebraically closed field, an n × n matrix A is nilpotent if and only if 0 is the only eigenvalue of A. Also, a square matrix that does not have two distinct eigenvalues must be the sum of a scalar matrix λI and a nilpotent matrix. Proof The second statement follows from the first because if λ is the only eigenvalue of A, then 0 is the only eigenvalue of A − λI (and A = λI + (A − λI) ). Suppose 0 is the only eigenvalue of A. Since the field is algebraically closed, the characteristic polynomial of A must be p(x) = xn. By the Cayley–Hamilton theorem, 0 = p(A) = An and so A is nilpotent. Note that the argument breaks down (not the Cayley–Hamilton theorem, which holds over any field) if the characteristic polynomial doesn’t factor completely. For instance, over the real field R, the matrix A = ⎡ ⎣ 0 0 0 0 0 −1 0 1 0 ⎤ ⎦ has zero as its only eigenvalue but is not nilpotent. (However A3 + A = 0, consistent with the Cayley–Hamilton theorem.) Now Mn(F) is not just a vector space under matrix addition and scalar multiplication, but also a ring with identity under matrix addition and
- 35. 8 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A multiplication, with scalar multiplication and matrix multiplication nicely intertwined by the law4 λ(AB) = (λA)B = A(λB) for all λ ∈ F and all A, B ∈ Mn(F). In this context, we refer to Mn(F) as an (associative) algebra over the field F. (The general definition of an “algebra over a commutative ring” is given in Definition 4.1.8.) By a subalgebra of Mn(F) we mean a subset B ⊆ Mn(F) that contains the identity matrix and is closed under scalar multiplication, matrix addition, and matrix multiplication (in other words, a subspace that is also a subring). Given a subset S ⊆ Mn(F), there is a unique smallest subalgebra of Mn(F) containing S, namely, the intersection of all subalgebras containing S. This is called the subalgebra generated by S, and is denoted by F[S]. In the case where S = {A1, A2, . . . , Ak} consists of a finite number k of matrices, we say that F[S] is k-generated (as an algebra) and write F[S] = F[A1, A2, . . . , Ak]. For a single matrix A ∈ Mn(F), clearly F[A] = {f (A) : f ∈ F[x]}. In fact {I, A, A2 , . . . , Am−1 } is a vector space basis for F[A] if Am is the first power that is linearly dependent on the earlier powers. Describing the members of F[A1, A2, . . . , Ak] when k 1, or even computing the dimension of this subalgebra, is in general an exceedingly difficult problem. Over an algebraically closed field F, a nonderogatory matrix is a square matrix A ∈ Mn(F), all of whose eigenspaces are 1-dimensional. This is not the same thing as A having n distinct eigenvalues, although the latter would certainly be sufficient. Nonderogatory matrices can be characterized in a number of ways, two of which are recorded in the next proposition. We postpone its proof until Proposition 3.2.4, by which time we will have collected enough ammunition to deal with it quickly. Proposition 1.1.2 The following are equivalent for an n × n matrix A over an algebraically closed field F: (1) A is nonderogatory. (2) dim F[A] = n. (3) The only matrices that commute with A are polynomials in A. Nowadays, the term “nonderogatory” often goes under the name 1-regular. The reason for this is that nonderogatory is the k = 1 case of a k-regular 4. This is equivalent to saying that left multiplying matrices by a fixed matrix A, and right multiplying matrices by a fixed matrix B, are linear transformations of Mn(F).
- 36. Background Linear Algebra 9 matrix A, by which we mean a matrix whose eigenspaces are at most k- dimensional. Later in the book we will be particularly interested in 2-regular matrices and, to a lesser extent, in 3-regular matrices. In a section on the most basic notions of linear algebra, it would be remiss of the authors not to mention elementary row operations, and their role in finding a basis for the null space of a matrix, for example.5 One should never underestimate the importance of being able to do row operations systematically, accurately, and quickly. They are the “calculus” of linear algebra. One should have the same facility with them as in differentiating and integrating elementary functions in the other Calculus. Recall that there are three types of elementary row operations: (1) row swaps, (2) adding a multiple of one row to another (different) row, and (3) multiplying a row by a nonzero scalar. We denote the corresponding elementary matrices that produce these row operations, under left multiplication, respectively by Eij (the identity matrix with rows i and j swapped), Eij(c) (the identity matrix with c times its row j added to row i), and Ei(c) (the identity matrix with row i multiplied by the nonzero c). We will also have occasion to employ elementary column operations that correspond to right multiplication by elementary matrices. Note, however, that right multiplication by our above Eij(c) adds c times column i to column j, not column j to column i. Here is a simple example to remind us of the computations involved in elementary row operations. In this one example, to encourage good habits, we will actually label each row operation using the lowercase version of the corresponding elementary matrix (e.g., e35 swaps rows 3 and 5, e21(−4) adds −4 times row 1 to row 2, and e4(2 3 ) multiplies row 4 by 2 3 ). We won’t spell that out in later uses (in fact, later eij will be reserved for something different—the “matrix unit” having a 1 in the (i, j) position and 0’s elsewhere). Example 1.1.3 Finding a basis for the null space of the matrix A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 0 2 −1 2 4 0 4 −2 1 2 3 −1 8 −1 −2 2 −4 7 3 6 0 6 −3 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 5. Repeated use of this procedure is really the key to computing the Weyr form of a matrix, as we shall see in Chapter 2.
- 37. 10 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A amounts to solving the homogeneous linear system Ax = 0, which in turn can be solved by putting A in (reduced) row-echelon form: A −→ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 0 2 −1 0 0 0 0 0 0 0 3 −3 9 0 0 2 −2 6 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ using e21(−2) , e31(−1) , e41(1) , e51(−3) −→ e23 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 0 2 −1 0 0 3 −3 9 0 0 0 0 0 0 0 2 −2 6 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −→ e2(1 3) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 0 2 −1 0 0 1 −1 3 0 0 0 0 0 0 0 2 −2 6 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −→ e42(−2) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 0 2 −1 0 0 1 −1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ The final matrix is the reduced row-echelon form of A, with its leading 1’s in columns 1 and 3. Hence, in terms of the solution vector x = ⎡ ⎢ ⎢ ⎢ ⎣ x1 x2 x3 x4 x5 ⎤ ⎥ ⎥ ⎥ ⎦ of Ax = 0, we see that x2, x4, x5 are the free variables (which can be assigned any values) and x1, x3 are the leading variables (whose values are determined by
- 38. Background Linear Algebra 11 the assigned free values). When we separate out the two classes of variables, the reduced row-echelon matrix gives us the equivalent linear system x1 = − 2x2 − 2x4 + x5 x2 = x2 x3 = x4 − 3x5 x4 = x4 x5 = x5 Recasting these equations using column vectors, we get ⎡ ⎢ ⎢ ⎢ ⎣ x1 x2 x3 x4 x5 ⎤ ⎥ ⎥ ⎥ ⎦ = x2 ⎡ ⎢ ⎢ ⎢ ⎣ −2 1 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎦ + x4 ⎡ ⎢ ⎢ ⎢ ⎣ −2 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎦ + x5 ⎡ ⎢ ⎢ ⎢ ⎣ 1 0 −3 0 1 ⎤ ⎥ ⎥ ⎥ ⎦ . Expressed another way, ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ⎡ ⎢ ⎢ ⎢ ⎣ −2 1 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎣ −2 0 1 1 0 ⎤ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎣ 1 0 −3 0 1 ⎤ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ is a basis for the null space of A. 1.2 BLOCKED MATRICES Staring at very large matrices can give one a headache, especially if the matrices require some sort of analysis under algebraic operations. So we should always be on the lookout for patterns, inductive arguments, and shortcuts. From a purely numerical analysis point of view, sparseness (lots of zeros) is often enough. But we are after something different that applies to even sparse matrices—the notion of “blocking” a matrix. It is a most useful tool. One can get by without much of an understanding of blocking in the case of the Jordan form. But the reader is warned that an appreciation of blocked matrices is indispensable for a full understanding of our Weyr form. There is not a lot to this. However, for whatever reason, blocking of matrices doesn’t seem to come naturally to some (even seasoned) mathematicians. Of course, every applied linear algebraist knows this stuff inside and out.6 6. The authors do not regard themselves as specialists in applied linear algebra.
- 39. 12 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A To keep the discussion simple, we will work with square matrices A over an arbitrary field. We can partition the matrix A by choosing some horizontal partitioning of the rows and, independently, some vertical partitioning of the columns. For instance, A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ combines the horizontal partitioning 6 = 3 + 1 + 1 + 1 with the vertical partitioning 6 = 1 + 1 + 3 + 1. This is a perfectly legitimate operation and very useful in some circumstances. But this particular partitioning of A is not a blocking in the sense we use the term, because if we have another 6 × 6 matrix B partitionedthesameway,wehavenoadditionalinsightintohowtocomputethe product AB. Blocking of a matrix comes when we choose the same partitioning for the columns as for the rows. For instance, using the same A, we could choose the horizontal and vertical partitioning 6 = n = n1 + n2 + n3 + n4 = 2 + 2 + 1 + 1 to give: A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A11 A12 A13 A14 A21 A22 A23 A24 A31 A32 A33 A34 A41 A42 A43 A44 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = (Aij), where the Aij are the ni × nj submatrices given by the rectangular partitioning. For example, A12 = 1 0 0 1 , A23 = 1 0 , A34 = 1 . In this context, where the same partition is used for both the rows and the columns, A is referred to as a block or blocked matrix and each Aij as its (i, j)th block. Note that the diagonal blocks Aii are all square submatrices.
- 40. Background Linear Algebra 13 Now given another 6 × 6 matrix B blocked in the same way (using the same partition), there is additional insight into how to compute the product AB. For instance, if B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 2 3 1 1 5 3 4 1 1 0 2 7 1 3 3 2 1 1 1 8 2 3 4 9 5 6 1 2 7 6 0 1 8 1 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ B11 B12 B13 B14 B21 B22 B23 B24 B31 B32 B33 B34 B41 B42 B43 B44 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = (Bij), then AB = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 7 1 3 3 2 1 1 1 8 2 3 4 9 5 6 1 2 7 0 0 0 0 0 0 6 0 1 8 1 2 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . Of course, one can get this by multiplying the 6×6 matrices in the usual way. Or one can multiply the pair of blocked matrices A = (Aij), B = (Bij), by the usual rule of matrix multiplication for 4 × 4 matrices, but viewing the entries of the new matrices as themselves matrices (the Aij, Bij) of various sizes. Since we have partitioned the rows and columns the same way, the internal matrix calculations for the product will involve matrices of compatible size. For instance, the (1, 2) block entry of each of the blocked matrices is an ordinary 2 × 2 matrix. In the product AB, the (1, 2) block entry becomes 4 k=1 A1kBk2 = A11B12 + A12B22 + A13B32 + A14B42 = 0 0 0 0 3 1 1 1 + 1 0 0 1 3 3 8 2 + 0 0 6 1 + 0 0 1 8 = 3 3 8 2 .
- 41. 14 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A Having done this for our particular B, one can spot the pattern in AB for any B, for this fixed A. (What is it?) But it requires the blocked matrix view to see this pattern in its clearest form. Of course, one can justify the multiplication of blocked matrices in general (those sharing the same blocking), without getting into a subscript frenzy. Our reader can look at the Horn and Johnson text Matrix Analysis, or the article by Reams in the Handbook of Linear Algebra, for more general discussions on matrix partitioning. Notice that in specifying the block structure of a blocked matrix A = (Aij), we need only specify the sizes of the (square) diagonal blocks Aii, because the (i, j) block Aij must be ni × nj where ni and nj are the ith and jth diagonal block sizes, respectively. Moreover, as will nearly always be the case with our blocked matrices, if the diagonal blocks have decreasing size, the whole block structure of an n × n matrix can be specified uniquely simply by a partition n1 + n2 + · · · + nr = n of n with n1 ≥ n2 ≥ · · · ≥ nr ≥ 1. The simplest picture occurs when n1 = n2 = · · · = nr = d, because blocking an n × n matrix this way just amounts to viewing it as an r × r matrix over the ring Md(F) of d × d matrices. If A = (Aij) is a blocked matrix in which the Aij = 0 for i j, that is, all the blocks below the diagonal are zero, then A is said to be block upper triangular. It should be clear to the reader what we mean by strictly block upper triangular and (strictly) block lower triangular. Our example A above is block upper triangular. We can (and will) simplify the picture for a block upper triangular matrix by leaving the lower (zero) blocks blank, so that, for our example we have A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . The reader may not know it (nor is expected to at this stage), but she or he is looking at the 6 × 6 nilpotent Weyr matrix of Weyr structure (2, 2, 1, 1). The point we wish to make is that our first partitioning of the same matrix is not as revealing as this blocked form.
- 42. Background Linear Algebra 15 Just as with ordinary matrices, the simplest blocked matrices A are the block diagonal matrices—all the off-diagonal blocks are zero: A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ A1 A2 ... Ar ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ . In this case, we write A = diag(A1, A2, . . . , Ar) and say A is a direct sum of the matrices A1, A2, . . . , Ar. If B = diag(B1, B2, . . . , Br) is a second block diagonal matrix (for the same blocking), then AB = diag(A1B1, A2B2, . . . , ArBr). Of course, sums and scalar multiples behave similarly, so our knowledge of a block diagonal matrix is as good as our knowledge of its individual diagonal blocks. This is a simple but fundamental observation, used again and again in canonical forms, for instance. Those with a ring theory background may prefer to view this as saying the following. For matrices blocked according to a fixed partition n = n1 + n2 + · · · + nr, the mapping θ : (A1, A2, . . . , Ar) −→ diag(A1, A2, . . . , Ar) is an algebra isomorphism (1-1 correspondence preserving addition, multipli- cation, and scalar multiplication) of the direct product r i=1 Mni (F) of the matrix algebras Mni (F) onto the algebra of n × n block diagonal matrices (with the specified blocking). We finish our discussion of blocked matrices with another seemingly trivial, but very useful, observation on block upper triangular matrices. The yet-to-be- described Weyr form (when in company with some other commuting matrices) is particularly amenable to this result, more so than the Jordan form. We state the result for 2 × 2 block upper triangular matrices, but there is an obvious extension to general block upper triangular ones. Proposition 1.2.1 Let m and n be positive integers with m n. Let T be the algebra of all n × n matrices A that are block upper triangular with respect to the partition
- 43. 16 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A n = m + (n − m): A = P Q 0 R , where P is m × m, Q is m × (n − m), and R is (n − m) × (n − m). Then the projection η : T −→ Mm(F), A −→ P onto the top left corner is an algebra homomorphism (that is, preserves addition, multiplication, and scalar multiplication) of T onto the algebra Mm(F) of m × m matrices. Proof Clearly η preserves addition and scalar multiplication. Now let A = P Q 0 R , A = P Q 0 R be in T . Since AA = PP PQ + QR 0 RR , we have η(AA ) = PP = η(A)η(A ). Thus, η is an algebra homo- morphism. Remarks 1.2.2 (1) Projecting onto the bottom right corner is also a homomorphism. (2) Also, if T is the algebra of block upper triangular matrices relative to the partition n = n1 + n2 + · · · + nr, then, for 1 ≤ i ≤ r, the projection onto the top left-hand i × i corner of blocks is an algebra homomorphism onto the algebra of block upper triangular matrices of size m = n1 + n2 + · · · + ni (relative to the implied truncated partition of m). This homomorphism is just the restriction of η in the proposition for the case m = n1 + n2 + · · · + ni.
- 44. Background Linear Algebra 17 1.3 CHANGE OF BASIS AND SIMILARITY Change of basis and similarity are really about reformulating a given linear algebra problem into an equivalent one that is easier to tackle. (It is a bit like using equivalent frames of reference in the theory of relativity.) These fundamental processes are reversible, so if we are able to answer the simpler question, we can return with a solution to the initial problem. Fix an n-dimensional vector space V and an (ordered) basis B = {v1, v2, . . . , vn} for V. The co-ordinate vector of v ∈ V relative to B is [v]B = ⎡ ⎢ ⎢ ⎢ ⎣ a1 a2 . . . an ⎤ ⎥ ⎥ ⎥ ⎦ , where the ai are the unique scalars for which v = a1v1 + a2v2 + · · · + anvn. If B is another basis, we let [B , B] denote the change of basis matrix, that is the n × n matrix whose columns are the co-ordinate vectors of the B basis vectors relative to B. This is an invertible matrix with [B , B]−1 = [B, B ]. Co-ordinate vectors now change according to the rule7 [v]B = [B, B ][v]B. Now let T : V → V be a linear transformation. Its matrix [T]B relative to the (ordered) basis B is defined as the n × n matrix whose columns are the co-ordinate vectors [T(v1)]B, [T(v2)]B, . . . , [T(vn)]B of the images of the B basis vectors. The reason why we work with columns8 rather than rows here is that our transformations act on the left of vectors, and our composition of two transformations is in accordance with this: (ST)(v) = S(T(v)). The correspondence v → [v]B is a vector space isomorphism from V to n-space Fn . What T is doing, under this identification, is simply left multiplying column vectors by the matrix [T]B: [T(v)]B = [T]B[v]B. (So, abstractly, a linear transformation is just left multiplication of column vectors by a matrix.) 7. A good way to remember this and other change of basis results is that primed and unprimed basis labels alternate. 8. This is the sensible rule, but unfortunately not all authors observe it. To break it invites trouble in the more general setting of vector spaces over division rings. (In that setting, one should also place the scalars on the right of vectors.)
- 45. 18 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A A permutation matrix is a square n × n matrix P whose rows (resp. columns) are a permutation of the rows (resp. columns) of the identity matrix I under some permutation p ∈ Sn (resp. p−1 ∈ Sn). (Here, Sn is the symmetric group of all permutations of 1, 2, . . . , n.) In terms of the matrix of a linear transformation, and in the case of a row permutation p, we have P = [T]B where B = {v1, v2, . . . , vn} is the standard basis of Fn and T : Fn → Fn is the linear transformation whose action on B is T(vi) = vp(i). For fixed V and basis B, the correspondence T → [T]B provides the fundamental isomorphism between the algebra L(V) of all linear transformations of V (to itself) and the algebra Mn(F) of all n × n matrices over F: it is a 1-1 correspondence that preserves sums, products9 and scalar multiples. The result should be etched in the mind of every serious student of linear algebra.10 Two square n × n matrices A and B are called similar if B = C−1 AC for some invertible matrix C. “Similar” is an understatement here, because A and B will haveidenticalalgebraicproperties. (Inparticular,similarmatriceshavethesame eigenvalues, determinant, rank, trace,11 and so on.) This is because for a fixed invertible C, and a variable matrix A, the conjugation mapping A → C−1 AC is an algebra automorphism of Mn(F) (a 1-1 correspondence preserving sums, products, and scalar multiples).12 And under an automorphism (or isomorphism), an element and its image have the same algebraic properties.13 This view of similarity is entirely analogous to, for example, conjugation in grouptheory.Butwhatisnewinthelinearalgebrasettingishownicelysimilarity relatestothematricesofalineartransformationT : V → V ofann-dimensional 9. If we had put the co-ordinate vectors [T(vi)]B as rows of the representing matrix, the correspondence would reverse products. 10. Unfortunately, nowadays some otherwise very good students come away from linear algebra courses without ever having seen this. 11. The trace, tr A, of a square matrix A is the sum of its diagonal entries. 12. The so-called Skolem–Noether theorem of ring theory tells us that these conjugations are the only algebra automorphisms of Mn(F). (See Jacobson’s Basic Algebra II, p. 222.) 13. Thinking of complex conjugation as an automorphism of C, we see that a complex number and its conjugate are algebraically indistinguishable. In particular, there is really no such thing as “the” (natural) complex number i satisfying i2 = −1, short of arbitrarily nominating one of the two roots (because the two solutions are conjugates). This is unlike the distinction between, say, the two square roots of 2 in R. Here, one root is positive, hence expressible as a square of a real number; the other is not. So the two can be distinguished by an algebraic property.
- 46. Background Linear Algebra 19 space under a change of basis from B to B . They are always similar: [T]B = C−1 [T]B C where C = [B , B]. Moreover, every pair of similar matrices can be viewed as the matrices of a single transformation relative to suitable bases. Ausefulobservationinthecasethat C isapermutationmatrix,corresponding to some permutation p ∈ Sn, but this time via the action of p on the columns of I, is that C−1 AC is the matrix obtained by first permuting the columns of A under p, and then permuting the rows of the resulting matrix by the same permutation p. For instance, if p = (1 2 3) is the cyclic permutation, then C = ⎡ ⎣ 0 1 0 0 0 1 1 0 0 ⎤ ⎦ and C−1 ⎡ ⎣ a b c d e f g h i ⎤ ⎦ C = ⎡ ⎣ i g h c a b f d e ⎤ ⎦ . A standard way of utilizing the transformation view of similarity, but with a matrix outcome in mind, is this: suppose we are presented with an n × n matrix A over the field F and we are looking for a simpler matrix B (perhaps diagonal) to which A is similar. Firstly, let V = Fn , let B be the standard basis for V, and let T : V → V be the linear transformation that left multiplies column vectors by A: T ⎛ ⎜ ⎜ ⎜ ⎝ ⎡ ⎢ ⎢ ⎢ ⎣ a1 a2 . . . an ⎤ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎠ = A ⎡ ⎢ ⎢ ⎢ ⎣ a1 a2 . . . an ⎤ ⎥ ⎥ ⎥ ⎦ . Note [T]B = A. Secondly, “using one’s wits” (depending on additional information about A), find another basis B relative to which the matrix B = [T]B looks nice. Thirdly, let C = [B , B] be the change of basis matrix. Note that C has the B basis vectors as its columns and is invertible. Now we have our similarity B = C−1 AC by the change of basis result for the matrices of a transformation. Again, suppose T : V → V is a linear transformation of an n-dimensional space. A subspace U of V is said to be invariant under T if T(U) ⊆ U (T maps vectors of U into U). Notice that a nonzero vector v ∈ V is an eigenvector of T (for some eigenvalue) precisely when v is invariant under T. (This provides a clear geometric picture of why a proper rotation of the real plane
- 47. 20 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A about the origin through less than 180 degrees can’t have any eigenvalues—no lines through the origin are invariant under the rotation.) If we choose a basis B1 for an invariant subspace U and extend it to a basis B of V, then the matrix of T relative to B is block upper triangular in which the top left block is m × m, where m = dim U, and the bottom right block is (n − m) × (n − m) : [T]B = P Q 0 R , where P is the matrix of T|U : U → U relative to B1. This observation can often be used as an inductive tool. (It also allows a neat noninductive proof of the Cayley–Hamilton theorem in terms of transformations, by fixing v ∈ V and taking U to be the subspace spanned by all the Ti (v). Then through a natural choice for B1, the matrix P is a “companion matrix” whose characteristic polynomial p(x) ∈ Mm(F) is easily calculated, and for which p(T)(v) = 0 is easily verified. The reader is invited to complete the argument, or to curse the authors for not doing so!) The kernel and image of a transformation T are always subspaces invariant under T. We record the following simple generalization. Proposition 1.3.1 Suppose S and T are commuting linear transformations of a vector space V. Then the kernel and image of S are subspaces which are invariant under T. Proof Let U = ker S. For u ∈ U, we have S(T(u)) = (ST)(u) = (TS)(u) (by commutativity) = T(S(u)) = T(0) = 0 , which shows T(u) ∈ U. Thus, U is invariant under T. Similarly, so is S(V). A vector space V is a direct sum of subspaces U1, U2, . . . , Uk, written V = U1 ⊕ U2 ⊕ · · · ⊕ Uk, if every v ∈ V can be written uniquely as v = u1 + u2 + · · · + uk, where each ui ∈ Ui. In this case, a union of linearly independent subsets from each of the Ui
- 48. Background Linear Algebra 21 remains linearly independent. Consequently, dim V = dim U1 + dim U2 + · · · + dim Uk. As with (internal) direct sums or products of other algebraic structures, one can verify that a sum U1 + U2 + · · · + Uk of subspaces is a direct sum, meaning U1 + U2 + · · · + Uk = U1 ⊕ U2 ⊕ · · · ⊕ Uk, by repeated use of the condition that for k = 2, directness means that U1 ∩ U2 = 0. In general, we check the “triangular conditions”: U1 ∩ U2 = 0 , (U1 + U2) ∩ U3 = 0 , (U1 + U2 + U3) ∩ U4 = 0 , . . . (U1 + U2 + U3 + · · · + Uk−1) ∩ Uk = 0 . An especially useful observation (when teamed with results for change of basis matrices) is the following. Proposition 1.3.2 Suppose T : V → V is a linear transformation and V = U1 ⊕ U2 ⊕ · · · ⊕ Uk is a direct sum decomposition of V into T-invariant subspaces U1, U2, . . . , Uk. Pick a basis Bi for each Ui and let B = B1 ∪ B2 ∪ · · · ∪ Bk. Then relative to the basis B for V, the matrix of T is the block diagonal matrix [T]B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ A1 A2 ... Ak ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , where Ai is the matrix relative to Bi of the restriction of T to Ui. Proof There is nothing to this if (1) we have a clear mental picture of what the matrix of a transformation relative to a specified basis looks like,14 and (2) appreciate that the restriction of a linear transformation T to an invariant subspace U is a 14. If one is constantly referring back to the definition of the matrix of a transformation, and consulting with the “subscript doctor,” this distraction may hamper progress in later chapters.
- 49. 22 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A linear transformation of U as a vector space in its own right. For instance, suppose dim U1 = 3 and dim U2 = 2 and we label the basis vectors by B1 = {v1, v2, v3} and B2 = {v4, v5}. Since U1 is T-invariant, for i = 1, 2, 3, the T(vi) are linear combinations of only v1, v2, v3, so in the matrix [T]B, the first three columns have zeros past row three. Similarly, for i = 4, 5, the T(vi) are linear combinations of only v4, v5, so in the matrix [T]B, columns four and five have no nonzero entries outside of rows four and five. And so on. 1.4 DIAGONALIZATION There isn’t a question that one can’t immediately answer about a diagonal matrix D = diag(d1, d2, . . . , dn) = ⎡ ⎢ ⎢ ⎢ ⎣ d1 0 · · · 0 0 d2 0 . . . ... 0 0 · · · dn ⎤ ⎥ ⎥ ⎥ ⎦ . For instance, its kth power is diag(dk 1, dk 2, . . . , dk n). So it is of interest to know when a square n × n matrix A is similar to a diagonal matrix. (Then, for example, its powers can also be computed.) Such a matrix A is called diagonalizable: there exists an invertible matrix C such that C−1 AC is diagonal. Standard texts include many interesting applications of diagonalizable matrices, from Markov processes, to finding principal axes of quadratic forms, through to solving systems of first order linear differential equations. Later, in Chapter 6, we examine an “approximate” version of diagonalization, which has modern relevance to phylogenetics and multivariate interpolation. Conceptually, the key to understanding diagonalization is through linear transformations T : V → V. The matrix of T relative to a basis B is diagonal precisely when the basis vectors are eigenvectors for various eigenvalues. In that case, the matrix is simply the diagonal matrix of the matching eigenvalues, in the order the basis vectors happen to be presented. An individual eigenvalue will appear on the diagonal according to its algebraic multiplicity. (This is just Proposition 1.3.2 when all the Ui are one-dimensional.) Sensibly, one should reorder the basis vectors to group together those sharing the same eigenvalue. To connect all this with matrices, we just use change of basis results. Presented with a small n × n matrix A, whose eigenvalues we know, we can test if A is diagonalizable by checking if the geometric multiplicities of its various eigenvalues sum to n. And an explicit C that diagonalizes A can also be found. Here is an example to remind us of the process.
- 50. Background Linear Algebra 23 Example 1.4.1 Suppose we wish to diagonalize the real matrix A = ⎡ ⎣ 3 1 1 1 3 1 1 1 3 ⎤ ⎦ . Using a first row cofactor expansion, we see that the characteristic polynomial p(x) of A is p(x) = det ⎡ ⎣ x − 3 −1 −1 −1 x − 3 −1 −1 −1 x − 3 ⎤ ⎦ = (x − 3)[(x − 3)2 − 1] + 1(−x + 3 − 1) − 1(1 + x − 3) = (x − 2)2 (x − 5). Hence the eigenvalues of A are 2, 5 with respective algebraic multiplicities 2 and 1. We need to check if these agree with the geometric multiplicities. We can compute a basis for the eigenspace E(2) using elementary row operations: 2I − A = ⎡ ⎣ −1 −1 −1 −1 −1 −1 −1 −1 −1 ⎤ ⎦ −→ ⎡ ⎣ 1 1 1 0 0 0 0 0 0 ⎤ ⎦ . The corresponding homogeneous system (2I − A)x = 0 has two free variables, from which we can pick out the basis B2 = ⎧ ⎨ ⎩ ⎡ ⎣ −1 1 0 ⎤ ⎦ , ⎡ ⎣ −1 0 1 ⎤ ⎦ ⎫ ⎬ ⎭ for E(2). For E(5) we proceed similarly: 5I−A = ⎡ ⎣ 2 −1 −1 −1 2 −1 −1 −1 2 ⎤ ⎦ −→ ⎡ ⎣ 1 1 −2 −1 2 −1 2 −1 −1 ⎤ ⎦ −→ ⎡ ⎣ 1 1 −2 0 3 −3 0 −3 3 ⎤ ⎦ −→ ⎡ ⎣ 1 1 −2 0 1 −1 0 0 0 ⎤ ⎦ −→ ⎡ ⎣ 1 0 −1 0 1 −1 0 0 0 ⎤ ⎦.
- 51. 24 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A This gives one free variable in the homogeneous system (5I − A)x = 0, from which we get the basis B5 = ⎧ ⎨ ⎩ ⎡ ⎣ 1 1 1 ⎤ ⎦ ⎫ ⎬ ⎭ for the eigenspace E(5). Thus, the geometric multiplicities of the eigenvalues 2 and 5 sum to n = 3, whence A is diagonalizable. We can diagonalize A explicitly with an invertible matrix C as follows. Let B be the standard basis for V = F3 and note A is the matrix of its left multiplication map of V relative to B. Next, form the basis for V B = B2 ∪ B5 = ⎧ ⎨ ⎩ ⎡ ⎣ −1 1 0 ⎤ ⎦ , ⎡ ⎣ −1 0 1 ⎤ ⎦ , ⎡ ⎣ 1 1 1 ⎤ ⎦ ⎫ ⎬ ⎭ of eigenvectors of A. Finally take these basis vectors as the columns of the matrix C = ⎡ ⎣ −1 −1 1 1 0 1 0 1 1 ⎤ ⎦ . The outcome, by the change of basis result for a linear transformation (looking at the left multiplication map by A relative to B and noting that C = [B , B]), is C−1 AC = ⎡ ⎣ 2 0 0 0 2 0 0 0 5 ⎤ ⎦ , a diagonal matrix having the eigenvalues 2 and 5 on the diagonal and repeated according to their algebraic multiplicities. In the above example, the diagonalization works over any field F whose characteristic is not 3. (When F has characteristic 3, the above A has 2 as its only eigenvalue but this has geometric multiplicity only 2, less than 3.) So, in general, diagonalization depends on the base field. For instance, real symmetric and complex hermitian matrices are always diagonalizable (in fact by an orthogonal and unitary matrix, respectively), but over the two element field, only the idempotent matrices E (those satisfying E2 = E) are diagonalizable. A frequently used observation is that an n × n matrix that has n distinct eigenvalues is diagonalizable. The general theorem is that A ∈ Mn(F) is diagonalizable if and only if the minimal polynomial of A factors into distinct linear factors. The minimal polynomial is the unique monic polynomial m(x)
- 52. Background Linear Algebra 25 ofleastdegreesuchthatm(A) = 0.Itcanbecalculatedbyfindingthefirstpower As of A that is linearly dependent on the earlier powers I, A, A2 , . . . , As−1 , say As = c0I + c1A + · · · + cs−1As−1 , and taking m(x) = xs − cs−1xs−1 − · · · − c1x − c0. The minimal polynomial divides all other polynomials that vanish at A. In particular, by the Cayley–Hamilton theorem, the minimal polynomial divides the characteristic polynomial, so the degree of m(x) is at most n. In fact, m(x) has the same zeros as the characteristic polynomial (the eigenvalues of A), only with smaller multiplicities. In some ways, the minimal polynomial is more revealing of the properties of a matrix than the characteristic polynomial. One can also show that, as an ideal of F[x], the kernel of the polynomial evaluation map f → f (A) has the minimal polynomial of A as its monic generator. This is as good a place as any to record another property of the minimal polynomial, which we use (sometimes implicitly) in later chapters. Proposition 1.4.2 The dimension of the subalgebra F[A] generated by a square matrix A ∈ Mn(F) agrees with the degree of the minimal polynomial m(x) of A. Proof Finite-dimensionality of Mn(F) guarantees some power of A is dependent on earlier powers, so there is a least such power As that is so dependent. Let (∗) As = c0I + c1A + · · · + cs−1As−1 be the corresponding dependence relation. Now I, A, A2, . . . , As−1 all lie in F[A] and are linearly independent by choice of s. We need only show they span F[A] in order to conclude they form a basis with s members, whence dim F[A] = s = deg(m(x)). In turn, since the powers of A span F[A], it is enough to get these powersaslinearcombinationsofI, A, A2, . . . , As−1.Butthisjustinvolvesrepeated applications of the relationship (∗): As+1 = AAs = A(c0I + c1A + · · · + cs−1As−1 ) = c0A + c1A2 + · · · + cs−1As = c0A + c1A2 + · · · + cs−2As−1 + cs−1(c0I + c1A + · · · + cs−1As−1 ) = cs−1c0I + (c0 + cs−1c1)A + · · · + (cs−2 + c2 s−1)As−1 and so on.
- 53. 26 A D V A N C E D T O P I C S I N L I N E A R A L G E B R A Unlike the minimal polynomial of an algebraic field element, minimal polynomials of matrices need not be irreducible. In fact, any monic polynomial of positive degree can be the minimal polynomial of a suitable matrix, and the same can happen for the characteristic polynomial.15 The following is the standard example: Example 1.4.3 Let f (x) = xn + cn−1xn−1 + · · · + c2x2 + c1x + c0 ∈ F[x] be a monic polynomial of degree n. Then the following “companion matrix” C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 · · · −c0 1 0 −c1 0 1 −c2 . . . ... . . . 1 0 −cn−2 0 0 · · · 0 1 −cn−1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ has f (x) as both its minimal and characteristic polynomials. To see this, one observes that the first n powers of C are independent so the minimal polynomial has degree n and necessarily agrees with the characteristic polynomial. On the other hand, the characteristic polynomial det(xI − C) can easily be computed to be f (x) directly, by a cofactor expansion in the first row (combined with induction for matrices of size n − 1 when evaluating the (1, 1) cofactor). Another useful point of view of diagonalizable matrices A ∈ Mn(F) is that they are precisely the matrices possessing a “spectral resolution”: A = λ1E1 + λ2E2 + · · · + λkEk where the λi are scalars and the Ei are orthogonal idempotent matrices, that is, E2 i = Ei and EiEj = 0 for i = j. (In the spectral resolution, there is no loss of generality in assuming that the λi are distinct, in which case the Ei are actually polynomials in A.) This is a nice “basis-free” approach. 15. Consequently, there can be no way of exactly computing the eigenvalues of a general matrix. Nor can there be a “formula” for the eigenvalues in terms of the rational operations of addition, multiplication, division, and extraction of mth roots on the entries of a general matrix of size bigger than 4 × 4. (This follows from Galois theory, more particularly the Abel–Ruffini theorem that quintic and higher degree polynomial equations are not “solvable by radicals.”) However, the matrices that arise in practice (e.g., tridiagonal) are often amenable to fast, high-precision, eigenvalue methods.
- 54. Background Linear Algebra 27 Generally, idempotent matrices play an important role in matrix theory. To within similarity, an idempotent matrix E looks like the diagonal matrix E = ⎡ ⎢ ⎢ ⎢ ⎣ 1 0 · · · 0 0 1 . . . ... 0 0 · · · 0 ⎤ ⎥ ⎥ ⎥ ⎦ , where the number of 1’s is the rank of E. In particular, idempotent matrices have only 0 and 1 as eigenvalues (combined with diagonalizability, this characterizes idempotents). Idempotent linear transformations T : V → V are exactly the projection maps T : U ⊕ W −→ U , u + w −→ u associated with direct sum decompositions V = U ⊕ W. Necessarily, U is the image of T, on which T acts as the identity transformation, and W is its kernel, on which T acts, of course, as the zero transformation. Since both these subspaces are T-invariant, a quick application of Proposition 1.3.2 gives the displayed idempotent matrix E of T for a suitable basis. In turn, by change of basis results, this justifies the above claim that idempotent matrices look just like E to within similarity. Again it is a transformation view that has led us to a nice matrix conclusion. 1.5 THE GENERALIZED EIGENSPACE DECOMPOSITION Recall that a matrix N is nilpotent if Nr = 0 for some positive integer r, and the least such r is called the nilpotency index of N. When our base field F is algebraically closed, many problems in linear algebra reduce to the case of nilpotent matrices. In particular, this is true in establishing the Jordan and Weyr canonical forms. The reduction is best achieved through the generalized eigenspace decomposition, which we will describe in this section. Nice though they are, diagonalizable matrices, at least those occurring in practice, form only a small class of matrices.16 The analysis of a general matrix requires a canonical form such as the rational, Jordan, or Weyr form, which 16. The relative size of the class of diagonalizable matrices in Mn(F) depends, of course, on the base field F and the order n of the matrices. For example, when n = 2 and F is the two element field, 8 out of the 16 matrices are diagonalizable. Things don’t improve in M2(R). Here a randomly chosen matrix still has only a 50% chance of being diagonalizable. Of course in Mn(C) for any n, with probability 1 a randomly chosen matrix will be diagonalizable because its eigenvalues will be distinct. However, when the eigenvalues are known not to be distinct, the
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- 56. noticeable to the large schooners and cutters of which so much is heard. The principal form in America laid down for all yachts used to be a long flat floor with very small displacement, great beam with a centreboard—the immense beam giving great initial stability. Large as well as small yachts were built to this design, and much used to be heard about their remarkable speed. A few years ago, however, two or three small yachts, amongst them the little 'Delvin' 5-tonner, built by Mr. W. Fife, jun., were sent over, all of fairly large displacement. These, without exception, put the extinguisher on all the American small yachts, by beating them time after time. The reason of it was that the English-built yachts could drive through what broken water or sea disturbance they met with, while the 'skim-dishes' could do little against it. Since those days the Americans have very materially altered their model, and both large and small yachts have been given more power; vide the examples brought out to compete with our yachts for the 'America Cup,' and those to which the 'Minerva' has so lately shown her tail. Where, therefore, great speed is required, and there is no limitation to sail-carrying power, a large displacement vessel is the best type to choose. Some small-yacht racing men do not like to be always remaining in their home waters, but prefer to go round to the regattas at other ports, and try their luck against the small yachts that gather at these meetings. They live on board, and sail their yachts round the coast. To such the large bodied boat is a regular frigate. The head-room is good, no lack of space is wanted for a comfortable lie down, and the owner and two friends, with racing sails and all other yacht paraphernalia, can stow away in the main cabin as cosily as can be.
- 57. 'Minerva,' 23 tons. Designed by W. Fife, 1888. During the last six years yacht designers have been spending their time in perfecting a vessel to be rated by length and sail-area alone. Boats of large displacement and moderate length, with good sail- spread, limited so that the boats might be rated under their several classes, gradually, but surely, gave place to boats of greater length, smaller bodies, and a smaller sail-spread. It does not appear, from the opinions of many who have published their views, that there is at the present time any particular desire to have good accommodation in racing yachts. The owners of the greater number of the 5-raters do not live in them, and the owners of the 40-raters have been so accustomed to great head-room in their vessels, that now, when, instead of having 7 feet to 8 feet, they still find they can walk about in the cabins, no complaints are heard; but with the lessons that Mr. Herreshoff has been teaching, there is every reason to believe that we may live to see a 40-rater launched with about 3 to 4 feet depth of body under water, and then perhaps there may come a reaction, and a return may be made to a moderately large displacement. Up to the present time the 5-raters have been kept fairly large, and owing to their beam, as far as internal accommodation is concerned, have room enough and to spare; but the raters of 1893 were not nearly of such large displacement as the boats of two years before, and they are wonderful to look at outside. The fin-keel requires great depth if it is to be of any real use, and it is in this particular point that small yachts suffer. If a 5-rater is to sail in all waters, and go the round of the coast regattas, then her
- 58. draught should be limited; of course, if the sole intention of the owner is that his yacht is never to race in any other locality than his own home waters, then, if the home waters be the Clyde, or Windermere, or Kingstown, there is no reason why depth should not be unlimited. On the other hand, should the yacht be intended for a sea-going vessel, then a heavy draught of water is not altogether desirable. There are times when a 5-tonner or rater may be overtaken by bad weather while making a passage, and when a comfortable harbour under the lee would be a most acceptable refuge to make for. There are scores of snug little places round the coast where a small craft could lie peacefully enough, provided her draught of water allowed her to make use of any one of them. The average depth of water at these bays or harbours is about 6 feet at low- water spring tides. Hence no yacht or rater of 30 feet or under should have a draught of more than 6 feet. The writer remembers only too well an occasion when, after leaving Campbeltown, in Cantyre, for a northern port in Ireland, a north-westerly gale sprang up, bringing with it rain and a sea fog. The distance across from the Mull is not more than a few miles, but when his yacht made the land it was blowing so hard he had to run for the nearest shelter. Alas! when he sighted the little tidal harbour he was steering for, it was low water, and his yacht, which drew 7 feet 6 in., could not enter. He had to lie at two anchors outside in the Roads with some half-a- dozen coasters, expecting, with every shift of the wind, that the anchorage might become one on a lee shore. The 'Humming Bird,' in 1891, left the Solent for Queenstown. She is a 2½-rater. After leaving Land's End the weather, which had been more or less fine, changed, and the sea getting up, it was decided to take her into St. Ives Harbour. She unfortunately drew more than 6 feet; the consequence was, though only 25 feet on the water-line, she was compelled to take her chance and drop anchor in the bay outside, because there was only 6 feet of water in the harbour.
- 59. None know the value of a moderate draught of water better than those who have cruised or raced afar from home, and groped their way into all kinds of out-of-the-way bays and harbours in small craft. The yachtsman who builds for racing only, possesses the means, and is ready to launch a new yacht to his name every other year, should (if he be a sensible man and proposes to himself to sell the yachts he has no further use for) think of the requirements of the market and his ability to sell. Soon the yacht mart will be flooded with a number of cast-off 5-and 2½-raters, all with a draught of water which would limit their sale to only a few places. There are many living at the present moment who will remember the time when even the large yachts of 100 to 200 tons were never given more than 12 feet draught. This was done to enable them to enter tidal harbours, the greater number of which only have a depth of 15 feet at high-water neeps. But there is another argument in favour of not having too great a draught of water, and that is, it is not an element of speed, beyond helping the sail power; and the existence of yachts like the old 'Fiery Cross,' which only drew 8 feet and was a most successful winner, and of the Herreshoff boats, which do not draw so much as the English-built raters and are the cracks of the day, points the lesson that it is well to put a limit where a limit may be altogether an advantage. Great care is necessary in apportioning out beam, no matter whether the yacht is to be of large or small displacement; great beam in the case of a yacht of small displacement is only suitable for waters such as Long Island Sound, or long rolling seas, and is useless in heavy broken water like that met with in our channels; because it is a difficult matter, without weight, to drive through the seas. When great beam is given to a yacht of large displacement, she may be able to fight her way through the water, but it will not be at the greatest speed for the given length, since it was proved by the old Solent 30-ft. and 25-ft. classes that when beating through a head sea a yacht of the same length, but of small beam, such as the 'Currytush' and the late Lord Francis Cecil's little 3-tonner
- 60. 'Chittywee,' were able to travel faster through the water whenever it was a hard thrash to windward. The general opinion of those competent to judge is, that 3½ to 4½ beams to the length on L.W.L. is about the most advantageous proportion, some going even so far as to assert that three beams may be given; but, in dealing with small yachts, 5-raters and 5-tonners, as this chapter does, the writer believes that four beams to length is a good proportion to meet all kinds of weather with; and if 30 feet be the length 7 ft. 6 in. the beam, and 5 ft. 6 in. the draught, such proportions will be found to give quite sufficient scope to any designer in order that a remarkably fast weatherly little ship may be the result of his calculations. The height between the decks with a large displacement would give 4 ft. 6 in. to 5 feet head-room. Nothing has been said about the sail-area, which should not be taxed. The element sail-area appears to be the stumbling-block in the present rating rule. It is limited, and the consequence is the cart is put before the horse, and the hull is built to the sail-spread. Thus the hull is being minimised to carry the small area allotted to certain lengths. There have been so many raters built since the present rule came in that it would take too much space to mention them all with their several points, but there is this fact to notice, which backs up what has been said before, that South-country designed boats seem to do well in their own waters, while those brought out in the Clyde fare best there. When Clyde 5-raters have gone South, they have performed badly—though the 'Red Lancer' in 1893 proved the exception to the rule—and the Solent raters that have found their way up North have made but a poor show. Mr. Arthur Payne is the king of draughtsmen on the Solent, and his yachts, with those designed by Mr. Clayton, also a prince among naval architects, have all had their turn at winning prizes when they have been properly sailed. Mr. Payne's designs mostly favour a fair amount of displacement, and 'Alwida,' built by him for Lord Dunraven in 1890, is a very fine example of the kind of craft he can produce. The
- 61. workmanship is fit to compare with the very neatest cabinet work. The following year the beam was increased by some inches, the length underwent a drawing out, and at the same time the body was tucked up to decrease the displacement. The next movement, if it is possible to judge by the 2½-raters, will be to follow in the steps of Mr. Herreshoff—who speaks for himself in other chapters. The 'Cyane,' another of Mr. Payne's 5-raters and an improvement on the 'Alwida,' has few fittings below, but there is great height between decks, and if she were changed into a cruiser, she has enough room to make her everything that can be desired, without greatly decreasing her speed. To describe all the 5-raters sent out to do battle by those Northern champions, Messrs. Fife and G. L. Watson, would be equally out of place here. Their boats are too well known all over the world both for speed and beauty of design, and if there is a point peculiar to either of them that marks their vessels and makes their meetings interesting and exciting, it is that while Mr. Watson's are extra smart in topsail breezes, Messrs. Fife's yachts are specially good in strong winds. 'RED LANCER' 11 tons T.M., 5-rater (Capt. Sharman-Crawford). Designed by Fife of Fairlie, 1892.
- 62. In mentioning these well-known names, it would be impossible to forget a name which will always be linked with the year 1892—viz. Mr. J. H. Nicholson, jun., of the firm of Messrs. Nicholson Sons, Gosport, the successful designer of the 5-rater 'Dacia' and the 2½- rater 'Gareth.' His boats are unique, and though they partake of the canoe form, still it is the shape adopted by Mr. Nicholson for his keel, and the design itself, which brought his name so prominently forward during the season of 1892 as one of the most successful designers in England. The 5-rater 'Dacia,' which he designed and built in 1892 for Mr. H. R. Langrishe, and which now belongs to Lord Dudley, proved herself far superior in all weathers to the yachts of her rating in the South. Most of the raters were designed with a square stern above water, whatever their shape might have been below; but the 'Dacia' is counter-sterned, and carries her ribbands fair from stem to taffrail, as far as can be judged from a long- distance view when she was hauled up. Whatever her length may be on the L.W.L., it must with a large crew aboard be so considerably increased as to almost make her another boat. At all events, she is a fine specimen of the advanced type of rater, and is good in all weathers. The 'Natica' and 'Red Lancer,' 5-raters by Mr. Watson and Mr. Fife, jun., must not be passed over unmentioned. Both these yachts belong to Belfast, which is at present the home of 5-rater racing. In the Clyde, where 5-tonners and 5-raters were once the fashionable classes, there is now not a single representative. The 'Red Lancer' is a fin-keel shaped vessel with great angle of sternpost, from the heel of which to the stem-head the line is run in a very easy curve. She has a very long counter, more than a third of which is submerged; but she is very pretty as a design, and though not of large displacement, is very roomy both on deck and below. She was originally fitted with a centreboard, but as it was not considered of any material benefit to her, this was taken out and the hole in the keel filled up with lead. The 'Natica' has a spoon bow, and is one of Mr. Watson's prettiest models as far as the modern racer can be termed pretty. She has been very successful in the North, and as
- 63. great curiosity was felt regarding her capabilities when compared with the South-country boats, she sailed round, and met the 'Dacia' at Torquay Regatta, where the best of three matches were won by 'Dacia.' It would have been better, perhaps, had the matches taken place off Holyhead—vide the case of the 'Vril,' 'Camellia,' and 'Freda'; however, there is no reason to disparage them as not giving a true indication of the respective merits of both yachts. So many races come off, both on the Solent and on the Clyde, in numerical comparison with what took place a few years ago, that the owners of small yachts rarely care now to go far away from home on the chance of obtaining sport when it lies comfortably to hand; but it is a thing to be encouraged, and when yachts have proved themselves champions in any particular waters, a trysting place should be chosen for the little winners to meet and try conclusions. This would also make yachtsmen anxious to possess not merely a racing machine, but a boat capable of going from port to port with a certain amount of comfort to her crew. 'Natica.' Designed by G. L. Watson, 1892.
- 64. PRACTICAL HINTS Buying In choosing a yacht there is, as with most other occupations, a right and a wrong way of going about it. First of all, the size has to be determined upon; but this can soon be done by referring to the length of the purse out of which the funds for keeping the yacht in commission are to be supplied. Yachts are very much like houses, and it is quite possible to buy a yacht or a house for such an insignificantly small outlay that to all unconcerned in the bargain it will appear a ridiculously cheap purchase. But this might not really be the case, because, though the original outlay may have been small, if a large number of servants or hands are required to keep either the one or the other up, it would be dear at any price should money not be forthcoming to meet the annual expenditure. It is, therefore, necessary, before making a purchase, to look ahead at the probable annual cost. At a rough estimate it may be laid down that each extra hand required (this does not refer to those necessary on racing days) will cost at least 25l. per season. A skipper may for his wages, clothes, c., make a hole in any sum from 30l. to 100l. per annum. In a 5-tonner, or a yacht of 30 feet and under, provided she has a gaff-mainsail and not a lugsail, one hand will be quite crew sufficient, with the owner, to take her about. The writer worked a 10-tonner with one hand for two seasons without finding her too heavy, but the addition of a boy made all the difference in the comfort. The cost of sails, gear, and the many small items of equipment which have to be renewed from time to time, cannot or should not be treated as if such casual expenses could only come about in some dim vista of futurity; for where in the case of sails 60l. might see the fortunate owner of a racing 5 in possession of a brand-new suit, the man with a 20 would find that sum barely sufficient to supply his yacht with a new mainsail and topsail.
- 65. In making a purchase, it is as well if it be possible to find out what kind of a yachtsman the owner of the yacht for sale is—that is, if he is a man who has made yachts and yachting his sole hobby, and has therefore been in the habit of keeping his vessels in the best condition. It makes all the difference whether you purchase from such a man, or from one who, having extracted all the good out of his yacht's gear and sails, has placed her in the market rather than go to the expense of giving her a new fit-out. In the case of a 5- tonner the difference in price between the purchase money of vessels owned by the two men might be from 50l. to 80l. or 100l.; but then in the case of the one there will only be one expense, viz.— that of the purchase money, whereas with the other it might be difficult to say how much might be required as outlay before the yacht could be made ready for sea. The Clyde and Southampton are the best and most likely places to find yachts for sale which have been well kept up and cared for. Buying from a thorough yachtsman who is known to spare no expense on his yacht will mean an absence of all bitterness and wrath, whereas in making the purchase from the skinflint, until a small fortune has been paid away the new owner will find that he has no satisfaction. In buying a small yacht, in fact any yacht, unless the purchaser has met with a vessel that combines all his requirements, it is always the wisest plan for him to spend as little as possible the first season on his new purchase—of course it is taken for granted that her sails and gear are in thoroughly good order—in altering any of her fittings to suit his own private fads; for if he changes his mind about his yacht's points, or sees a vessel he may like better, he should remember that he must not expect to get his money back again when wanting to sell. By the end of the first season, he will most likely have found out whether he will keep the yacht, and therefore whether she really suits him, when he can do what he likes to her. It must be borne in mind, too, that the inside fittings of a yacht's cabin form the most expensive part of her hull; and alterations below always mean a goodly expenditure.
- 66. Avoid all yachts which are either coated outside or filled in at the garboards inside with cement, as water will leak in between the cement and skin, and rot must ensue. Fitting out. In fitting out, two very important points have to be thought of— viz., if the yacht is not coppered, what is the best paint to coat her with, and what is the best method of treating the decks? With regard to the first question, there are two paints which the writer has never yet seen used in the yachting world, except on his own boat, and which can be highly recommended. One is the black priming varnish used on iron ships, and especially in the Navy. He gave this, some years ago on the Clyde, four months' good trial. It was used on a boat kept out for winter work which lay in a little harbour well-known for its fouling propensities. At the end of the four months there was absolutely no growth or sign of weed of any kind. Where it is to be obtained he is unable to say, as the coat of paint that was put on his boat was given him by a naval officer. The other paint is called after the inventor, 'Harvey's Patent.' The writer's experience of this is as follows:—A friend sent him a tin to try, and to give his opinion upon. Accordingly his boat, which had been lying up Portsmouth Harbour some six months at her moorings, was brought down to Priddy's Hard and hauled up. She had, though coated with a very well-known patent, from 7 to 10 feet of weed floating astern of her at the time, which had to be removed. After being thoroughly cleaned, left to dry for a few days, and having her paint burnt off, a coat of priming was given, followed by two coats of the Harvey. The boat was then launched and towed back to her moorings, where she was left for over 20 months. At the end of that time she was hauled up, prior to being put into commission; and there was no sign of grass or weed; slime, with an almost imperceptible shell-fish growth, being all that was visible on her bottom. The boat was seen by a good many naval and other men during the time she was at her moorings, and they remarked on the quality of the paint. One great
- 67. point about the Harvey must be mentioned, and that is, it dries very quickly when put on. It is a good thing to warm it before using, as it is apt to get hard and soak up the oil; but it soon softens, and after being properly mixed works well. Before touching the decks, the spars and blocks will always require to have the old varnish of the past season scraped off them, and will have then to be re-varnished. In scraping the spars care should be taken that the knife, scraper, or glass be drawn with, and not against, the grain of the wood. The scraping will always be achieved with greater facility if the spar or block in hand is slightly damped, and the scraper or knife-blade employed has its edge turned over a little. This latter is done by drawing the side of the edge along the back of a knife or steel tool. After scraping, the whole spar should be rubbed down with sand-paper, prior to its receiving a coat of varnish. The brushes employed should be either well-used ones, or, if new, ought to be well soaked in water prior to use, as this will prevent the bristles falling out during the process of varnishing. Nothing is so provoking as to have to be continually picking out bristles from the varnish; of course, what holds good about varnishing holds good in the matter of painting. When using copal varnish, it is as well to pour out only as much as may be wanted for the time being into an old tin or jar, because it very soon hardens on exposure to the air, and then becomes useless. For the same reason the varnish bottle or can should never be left uncorked. Two coats of varnish thinly laid on ought to suffice at the beginning of the season, and a third coat may be given as the season progresses. With regard to the decks. Everything depends on the state of the decks themselves and how they are laid. If they are made of wide planking, which is rarely, if ever, the case when the workmanship is that of a yacht-builder, they should be painted; if, however, the decks are laid with narrow planking fined off with the deck curves at the bow and stern, then, notwithstanding the beauty of white decks, it is better to varnish them. Varnishing keeps them hard, and saves
- 68. many a heart pang when the little yacht is visited by a friend with nails in his boots or a lady in small heels. If the decks be worn at all, a coat of varnish is a capital thing. After trying decks varnished and unvarnished, experience confesses that the joys of beholding a white, spotless deck in a small yacht are more than outweighed by the sorrow and annoyance of seeing deep nail-marks imprinted on it. As decks, when cared for, are always varnished when a yacht is laid up for the winter, this varnish has necessarily to be removed prior to a start on a season's yachting. The best method by which this can be carried out is as follows:—Black ashes, Sooji Mooji, or one of the many preparations of caustic potash, should be procured from a ship-chandler, and mixed in an iron bucket with warm water in the proportion of one-third black ashes to two-thirds water, according to the strength required. As soon as the sun has set the mixture must be poured over the deck, which must be left well covered with it till an hour before sunrise. The mixture, which will have dried during the night, must now be treated with hot water and well rubbed into the varnish, and fresh buckets of water must be kept applied till every particle of the mixture with the varnish has been cleared off and out of the deck planking. If the mixture is applied or allowed to remain on the deck while the sun is up, it will be certain to eat into and burn it. There are two or three ways of laying decks. One is to have the planks nailed down to the beams, the nails countersunk, and the holes filled up with wood plugs to hide the nail-heads. This is generally done by men who have not had much to do with yacht- building. The common method employed is to drive the nails diagonally through the edge of the plank into the beam. Nails let in horizontally and driven into the next plank, two or three cotton threads having been placed between, keep the two planks in position. Each plank is similarly treated, and when all the planks have been fitted and jammed together, marine glue is poured into the seams. As soon as the glue has set and hardened the decks are planed, and finished off. The third method is not so pretty perhaps,
- 69. but is believed from practical experience to be the best. The planks are mortised together, varnished, and then brought tight up. The whole deck is often built and made ready to fit before it is put into position, so that when it is laid on the beams, all that is required is to nail it down into its place. The writer has had experience with the second kind of deck mentioned here in nearly all of his yachts, and of the third method of laying decks in the 'Cyprus.' She was about five years old when he bought her, and that is a good age for a racing 5-tonner's decks to last sound and without a leaky spot to be found anywhere. Her decks were certainly kept varnished, for the simple reason stated above, that visitors might be always welcome, no matter what description of foot-gear had been supplied to them by their bootmakers. It is not an uncommon practice to have a yacht recoppered, though her copper may be in good condition and even new. When such a proposition is made, which is not infrequently done by skippers wishing to play into the yacht-builder's hands, and thinking more of their own pockets than their master's interests, the yachtsman must remember that every time his yacht is coppered her skin is made more porous, and she herself heavier in the water, since the planking will naturally sodden with greater rapidity. If the incipient yachtsman has bought the hull and spars of a yacht that is only partially built or finished off, a few more hints must be added, which will give him food for reflection, and may prove of service. When a yacht likely to suit has been heard of, nothing being known of the owner, the next thing should be to try to discover whether she is sound or possesses any weak places. The purchaser should overhaul her outside just below the channels, and examine if the yacht has been frequently caulked between the seams of the planking, or if there are any signs of weeps of any kind about that part or elsewhere. The weeps will be shown most likely by a rusty discolouration. If the yacht is coppered, wrinkles must be looked for
- 70. under the channels, runners, and about the bilge. They will show if the yacht has been strained at all. A knife should next be taken, and the point driven into the planking about the water-line, where it joins the sternpost and stem, and then along the two lower garboard strakes, especially if cement has been used to fill in between the keel and planking, to discover if there is any sign of dry rot, sap rot, c. Inside, under the cabin floor, the timbers, deadwoods, and the garboard strakes if the yacht be coppered, should be tested in the same way. If the yacht has iron floors, these should be carefully examined for galvanic action or decay. The heads of the bolts which go through the lead keel should be scraped to see whether they are made of iron, metal composition, or copper. If they are iron or steel, most likely they will require to be renewed, because galvanic action is very soon set up between the lead and steel. Outside, copper shows wear and tear more quickly near the stem and sternpost and along the water-line. In the cabin itself the deck ceiling should be examined for weeps and leaks, especially about the bits forward and near the mast, also wherever a bolt-head is visible. On deck, a look round the covering board will discover whether it has been often recaulked, by the seam being extra wide. The heat of a stove below is frequently the cause of the deck forward leaking. The deck seams should not be wider there than at any other part. All the spars should be examined, and if there are no transverse cracks, longitudinal ones may be held of no consequence. The weak parts of the mast are generally to be found between the yoke and cap, where the eyes of the rigging rest. Rot is often found there, and strains are met with up the masthead. The boom shows its weakness at the outer end by small cracks, and the bowsprit by the gammon iron and stem-head. If the above rough survey proves all correct, attention must be given next to the rigging, sails, and gear. Wear in the wire rigging is shown by its being rusty, the strands stretched, or by the broken threads of a strand appearing here and there. If the jib, throat, peak halliards, and mainsheet are new, or have seen the work of one season only, they will not require much overhauling. With the other running rigging the strands should be untwisted, just enough to see whether the heart of the rope is fresh
- 71. and not rotten. The blocks ought to be of a light colour without cracks in them, and iron strapped inside. The sails will not show either mildew marks or discolouration if they are in good condition. The chain and anchors to be in good order should not be rusty, but clean and well galvanised. They should be looked at to discover whether they have ever been regalvanised. This will be noticed by the links presenting a rough, uneven surface, where there was rust or decay before the repetition of the process of galvanising. Sometimes at fitting-out time an owner finds that he has to provide his yacht with a new anchor. It may help him, therefore, in his choice if the writer gives his experience in the matter of ground tackle or mud-hooks. There are a number of patents in the market, the most patronised of which are Trotman's, Martin's, Smith's, and Thomas Nicholson's. All these have many good points, with a weak one here and there to keep the competition in anchor designing open to improvement. Trotman's anchor has movable arms and stock, stows away well, and is a fine holding anchor when once it bites; but it is often very slow at catching hold, and this is dangerous when the anchorage happens to be close and crowded, as, for instance, is frequently the case at Kingstown, Cowes, c., during regatta time. If the anchor does not catch at once on such occasions the yacht may drift some distance before she is brought up, and with little room this operation is performed, more often than not, by collision with some vessel astern. The Martin anchor and the Smith both work on a different principle from any of the others, in that their arms move together so as to allow both flukes to act at the same time. Of the two the Smith, which has no stock, is preferable for yacht work. The Martin has a stock which is fixed on the same plane with the arms. Both anchors catch quickly and hold well as long as the bottom is not rocky or very uneven, when they are apt to get tilted over and lose any hold they may have at first obtained. Their worst failing is that
- 72. of coming home under the following conditions. If the yacht yaws about, owing to strong tides, winds, or boisterous weather, the flukes of the anchor are prone, when working in their holes, to make them so large that they gradually meet each other and finally become one big hole; the anchor then invariably trips, comes home, and the yacht drags. On the other hand, the Smith and Martin anchors stow away better than any others, and when on deck lie flat and compact. The Smith anchor makes a capital kedge. Its holding power is so great that it is not necessary to carry one of anything like the weight that would be required were any other patent anchor employed. The great point in favour of Smith's over that of Martin's anchor is that, should it foul a mooring or warp, it can be easily tripped. The tripping is done by letting the bight of a bowline slip down the chain and anchor till it reaches the arms, and then hauling on it. The best of the patents, however, is an anchor that was brought out some years ago by Messrs. Thomas Nicholson, of Southampton and Gosport. It can be stowed away in a very small space, since the arms are removable. It is a quick catcher, and is, at the same time, very powerful and trustworthy. The arms stand out at the most effective angle for insuring strength of grip, while the shank is long, and, though light and neat- looking—it is flat-sided—has sufficient weight and substance in it to stand any ordinary crucial test. The flukes from their shape appear somewhat longer in proportion to their width than the usual patterns; but this arises from the sides being slightly bent back, with the object of making the fluke more penetrating, which it certainly is. The old fisherman's anchor with a movable stock is, after all, as good an anchor as any yachtsman need want. It is not a patent, and is accordingly very much less expensive. Should necessity ever compel the making of a small anchor, then the two great points
- 73. which it must possess are, length of shank (because greater will be the leverage), and the placing of the arms so that they do not make a less angle with the stock than, say, 53°. After a long practical experience with almost every kind of anchor, the writer believes that two good, old-pattern fisherman's anchors, with movable stocks (the movable stock was a Mr. Rogers' patent), are all that any yacht need require or her mud-hooks; but if it is thought fit to have patent anchors, then either a couple of Thomas Nicholson's anchors, or one of these and a Smith, ought to form the yacht's complement. All being satisfactory, if the yacht is a 5-rater the first thing to be done will be to have the lugsail altered into a gaff-mainsail for handiness sake. This will be only a small expense, since the great peak of the lugsail will allow of its head being squared. Very little if anything need be taken off the head of a high-peaked lugsail when the gaff employed is hinged on to the jaws, as such a gaff can be peaked with far greater ease and to a much greater extent than when fitted in the ordinary. The writer has employed the following method for fitting up the interior accommodation of a 5-ton yacht, and he can highly recommend it as most convenient, and at the same time handy to clear out either on a racing day or when about to lay the yacht up:—All woodwork, such as lockers or fore-and-aft boards (used for turning the sofas into lockers), should be fixed in their places by hooks, or at any rate by screws. Nothing should be a fixture except the two sofa-seats in the main cabin, the one forward of the mast, and the two sideboards fitted aft at each end of the sofas. If the yacht has to race, these sideboards should be made self-contained, and to shape, so that they may fit into their places and be kept there by hooks or catches. There should be only a curtain forward between the forecastle and main cabin, and instead of a regular solid bulkhead aft, gratings should take its place, with one wide grating as a door. This will keep the store room aft ventilated. If there is sufficient length to permit of transverse gratings about 20 inches apart and 2 feet high by the mast, as before explained when describing the 'Lorelei,' by all means let these form one of the fittings to hold the sail bags. In the locker astern of
- 74. the after bulkhead gratings, the skin should be protected by battens 2½ to 3 inches wide and from 1 to 2 inches apart. This will keep whatever is stowed there dry from any little weep or leakage that may occur in the planking. There should be no ceiling either in the main or fore cabin, and if battens are thought necessary to prevent damp getting to the beds when left folded up in the bed-frames, then three, or at the outside four, some 4 or 5 inches apart, should be screwed up just in the position where the shoulders of a sitter would be likely to rest against them. Four or five may be fitted up on each side of the forecastle. The upper batten should be higher up than the top one in the main cabin, as it may be useful for screwing hooks into. The writer, however, prefers in the main cabin, instead of any battens, clean pieces of duck, or, what is better still, Willesden cloth (waterproof), made to hang loosely from hooks, reaching down to the sofas, and cut to the shape of the after sideboards, holes being sewn in to allow the iron hooks which carry the bed-frames to come through. This fitting always lightens up the cabin, and is easily taken down and scrubbed. For beds, the iron frames supplied to all yachts' forecastles for the men, with canvas bottoms to them, are far the best and most comfortable. They take up less room than a hammock, and stow away nicely against the cabin's side when not in use. With these frames the writer has used quilted mattresses, the heads of which have ticking covers large enough to hold a pillow, and the whole is sewn on to strong American or waterproof cloth, which forms a covering when the bed and its blankets are rolled up and have to be stowed away. In the forecastle, a movable pantry may be screwed up against the battens on the port side (the bed will be on the starboard side). This should be an open case with three shelves and two drawers underneath. The upper shelf must be divided off to take the three sizes—dinner, soup, and small plates. Between the plates, outside the divisions, there can be uprights on which to thread double egg- cups. On the lower shelf there should be holes cut to carry tumblers,
- 75. and between the tumblers slots for wineglasses. The bottom shelf is for cups and saucers. One of the drawers ought to be lined with green baize to hold silver plate and knives. If the sideboards aft are fixtures, a tin case made to the shape of the yacht's side, to rest on the part of the sideboard on which the lid hinges, and reaching up to the deck, is a capital fitting to have. The inside should be arranged in partitions to hold tea, coffee, sugar, biscuit, and other square canisters, also Dutch square spirit bottles. The door may be double, or if single, should open from the bottom and trice up to a hook overhead, so that it may not in any way hinder the opening of the sideboard lid at the same time. Two or three movable shelves placed right in the eyes of the yacht forward make useful stowage room for a man to keep his clothes, as there they stand less chance of getting wet. Between the sideboards aft a removable box ought to be fixed with screws, of sufficient depth to hold an iron bucket, washing basin, and all the conveniences of a lavatory. This will be directly under the cabin hatch, and from 8 to 10 inches abaft it. The lid should leave a few inches space clear to receive it when opened back. Curtains made of duck or Willesden cloth, to hang down loose over the sideboards at each side to the depth of 6 inches, and hung from hooks in the deck above, will be found useful for keeping all stray splashes, that may fall inboard, from going on the sideboard lids, and thence among the dry goods and provisions stowed away in them. At the back of the lavatory box will be the after-grating and locker, and standing out from the grating, about 10 inches to a foot square, and 15 to 18 inches deep from the deck, there should be a cupboard, painted white inside, or, better still, lined with copper silver-plated to reflect the light, and a transparent spirit compass should then be fitted to hang through the deck above it. The brass rim for carrying the gimbles and binnacle lid outside must be screwed down to the deck on doubled india-rubber to prevent leakage. The cupboard door must have ventilating holes in it at the top and bottom, and a square hole to hold the lamp should be cut in the door between the upper and lower ventilators. On the opposite side from that on which the tin case is fixed, and coming out from the grating the same distance as the compass box, two bookshelves
- 76. can be fitted, which will prove most useful. On deck, the fittings and leads that are mentioned in the description of the 'Cyprus' cannot be improved upon, except that rigging screws are neater, and give less trouble than dead-eyes and lanyards, which have to be continually set up. Lanyards, however, give more life to a mast, though it may appear almost imperceptible, and by so doing ought to render it less liable to be carried away. The sliding lid of the companion hatch should padlock on to a transverse partition between the combings, and it is a good plan to have this partition on hinges, so that at night, when the hatch-cover is drawn over, the partition may lie on the deck and so leave an aperture for ventilation. The windows of the skylight will be all the better for being fixtures and should not open; if ventilation be required, the whole skylight can be taken off; this will prevent the leakage so common with hinged windows. A mainsheet horse and traveller with two quarter leading blocks are better than a double block shackled on to an eyebolt amidships, because a more direct up and down strain can be obtained when the boom is well in. In any yacht of 25 feet in length or under, the wisest plan to adopt with regard to a forehatch is to do away with it and only have a large screw deadlight; if a small deadlight be preferred, then it ought to be placed about 12 to 18 inches ahead of the bits, and a copper cowl, to screw into the deadlight frame, should form part of the fittings, for use when the yacht is laid up, in order to let air into and so ventilate the cabin. It is certainly a great advantage to have the spinnaker ready in the forecastle for sending up through a hatch, but as this is the only good reason why a hatch should be thought requisite in a small yacht, and since it is a fruitful source of leakage and danger, especially when, as is sometimes the case, the lid has not been fastened down and a sea sweeps it off the deck, it is better to abolish the fitting altogether. A small rail ahead of the mast, bolted through the deck and stayed to the mast below (in order to take off all weight from the deck and beams), and a rail abreast of the lee and weather rigging, should form all that is required for belaying halliards, purchases, tacks, c. In most of the 5- and 2½-
- 77. raters the halliard for the lugsail is led below the deck, and the purchase is worked by taking turns round a small mast-winch in the cabin. It is a great advantage to have a clear deck free from ropes, and it would be a saving of labour to have all a cutter's purchases led below to a winch. For a small yacht it is as well to have the jib, throat, and peak halliards of four-strand Manilla rope, but wire topsail halliards are a very decided improvement on hemp or Manilla. Wire has little or no stretch in it, and a topsail halliard is the last rope a seaman cares to disturb after it has once been belayed, it may be to lower and take in the sail. All purchases ought to be made of European hemp-rope, with the exception of that attached to the copper rod bobstay. All headsheets should lead aft and belay on cleats bolted on to the combing of the cockpit. It is becoming the custom to have all the bowsprit fittings fixtures. A steel or copper rod from the stem to the cranze iron at the bowsprit end serves as a bobstay, which, with the shrouds, are screwed up with rigging screws. No such thing as reefing, or bringing the useless outside weight of the spar inboard, is thought of by many racing men now-a-days. Fiddle-headed and spoon bows have introduced this fashion, but 14 to 16 feet of a 5½- inch spar is no trifle to have bobbing into seas, and making the boat uneasy, when half the length, or less, would be quite sufficient to carry all the jib that can be set. No bowsprit belonging to a straight- stemmed cutter should be a fixture, and the best and neatest fitting for the bobstay is a rod with a steel wire purchase at the end. The shrouds should be in two lengths of wire shackled together, as in topmast backstays, and, leading through the bulwark, should screw up to bolts in the deck especially formed to take a horizontal strain. Selvagee strops can be used for setting up the intermediate lengths. If the eyes of the rigging are covered with leather which has not been painted, then the bight of each eye ought to be left standing in a shallow dish of oil. The leather will thus soak itself, and the oiling will preserve it from perishing.
- 78. In sending up rigging it must always be remembered that the lengths of the port and starboard rigging are arranged so as to allow of the starboard fore rigging being placed into position first, then that to port, the starboard backstay rigging going up next, followed by that to port, after which the eye of the forestay will go over the masthead and will rest on the throat halliard eyebolt in the masthead. All block-hooks should be moused. A mousing is made by taking two or three turns of spunyarn round the neck and lip of the hook followed by a cross turn or two to finish off. This prevents the hook from becoming disengaged. In some yachts double topmasts and double forestays are used. The former are only fitted where the yacht carries two sizes of jib- topsail, one for reaching and the other for beating to windward. Whilst one is up, the other can be hooked on, so that no time need be lost in setting. A medium-sized sail, however, capable of being used for reaching or beating, is all that is really required. The shifting of two jib-topsails entails the presence for some time of one man at least forward on the bowsprit end, and the less the men are forward of the mast the better, if it is desired to get the best work out of a small yacht, and the yacht herself is in proper trim. There is more to be said, however, in favour of double forestays, since they allow of a foresail being sent up whilst another is already set and drawing, and the work is done inboard, while the difference between a working and a balloon foresail is far greater than in that of two jib- topsails. The writer has never used double forestays, but he believes so thoroughly in the foresail, as a sail, that he has always carried three—a working, reaching, and a balloon. He has the luff of each foresail fitted with loops at regular intervals, after the manner of gaiter lacings, otherwise called 'lacing on the bight.' These are made either of light wire or small roping. The upper loop reaches down to the next below it, so that the loop below may be passed through, and so on, till the tack is reached. When setting a foresail the upper loop is passed over the forestay before the lower one is threaded
- 79. through it, and so on with all the loops in turn. The tack has a single part, which, after it has been passed through the lowest loop, is made fast to the tack-downhaul. When shifting foresails, the sail is lowered, tack let go, and the lacing comes away by itself; then the new sail can be hooked on to the halliards and laced to the forestay as quickly as it can be hauled up. When the sea is smooth there may be no necessity for unlacing the working foresail should the shift have to be made from that sail, especially if it has soon to be called into use again. The above method will be found far superior to that of hanks, which are always getting out of order and not infrequently refuse to do their duty altogether. In mentioning the shifting of sails, there is one point to which nothing like sufficient attention is paid, and that is to the lead of sheets. Many a good jib has been destroyed and pulled out of shape through a bad lead, and more than one race has been lost through the bad lead of a reaching or balloon foresail sheet. When jibs or foresails are changed, the greatest care should be taken to see that the leads told off for their sheets are really fair—that is, that the pull on the sheet does not favour the foot more than the leach of the sail, or vice versâ. In the case of a balloon-foresail its sheet leads outside the lee rigging and belays somewhere aft. The man attending the sheet should take it as far aft as a direct strain will permit, and not belay it to the first cleat that comes to hand; otherwise the sail will simply prove a windbag taking the yacht to leeward rather than ahead. There is a fitting which must not be passed over that is now almost universally adopted on large yachts, but is equally important on small ones—that is, an iron horse at the main-boom end for the mainsail outhaul to travel on. It was originally invented by that most skilful helmsman Mr. W. Adams, of Greenock, to obviate a difficulty so common in square stern boats with booms stretching to n length over the transom. He fitted the boom of his little racing boat with a horse, which came from the boom end to within easy reach for unhooking the clew of the sail, and so saved the trouble of having to
- 80. use a dinghy for the purpose. The idea was soon taken up by Clyde yachtsmen, for it was found so much easier to get the mainsail out on the boom than with the traveller working on the boom itself. Whilst on a subject connected with mainsails, the writer can recommend for the gaff and head of the mainsail, instead of the ordinary long rope lacing commonly in use, separate stops or seizings to each eyelet-hole. The seizing can be done in half the time it takes to properly lace the head of the sail to the spar; it looks quite as well and does its work better. For fastening the luff of the mainsail to the mast-hoops, instead of seizings he has used hanks, and has found them very handy and neat. The hanks used are riveted on to the mast-hoops. He has now had them in constant use for over twelve years, and has never had occasion to find any fault whatever with them. In one yacht he kept two mainsails in use for cruising and racing, and thus preserved the racing mainsail in good condition for a considerably longer period than would otherwise have been the case, and with the fittings just named the shift of sails was a small matter. Topsails, perhaps, are the sails which require renewing more frequently than any other, as they get out of shape so quickly if very much is demanded from them. For a small yacht, if she carries a topmast, three topsails are a sufficient outfit. They should be a jibheader, a gaff, and a balloon or jackyarder. One yard ought to serve for both the gaff and jackyard topsail, and these sails should be made the same length on the head. This will save having to carry about a deckload of timber. OUTFIT It is frequently a question of great moment, what kind and what amount of outfit it is necessary to take away on a summer's cruise, and the writer finds it a great convenience to keep a list of everything that goes to form not only his sea kit, but stores and
- 81. necessaries as well. Such a list prevents one from forgetting small necessaries. A small air-tight 'uniform tin case' and a painted seaman's bag are the best equipment for carrying clothes. The lists are as follows:— FIRST LIST: THE KIT The tin case holds— A dress suit and shoes A shore-going suit 3 linen shirts 6 collars White ties Gloves Ink, blotting paper, paper and envelopes Mr. Lloyd's Euxesis As may be seen, the tin box only contains the shore-going outfit. The Euxesis mentioned is for those who shave, as with it there is no need of hot water to perform the operation. In the bag should be— 4 flannel shirts 2 pairs of flannel pygamas ½ doz. pairs of socks, 2 pairs of which should be thick 2 pairs of thick warm stockings 1 pair of warm slippers 2 pairs of common blue india-rubber solid shoes 1 pair of brown leather shoes 2 blue guernseys, hand knit 4 bath towels and ½ doz. others Sponge bag
- 82. Dressing case 1 suit of thick pilot cloth 1 old pair of thick blue trousers 1 large thick square comforter 1 common serge suit 1 pair of mittens 1 pair of tanned leather boots For comfort in a small yacht it is impossible to do with less. Of course it may be thought foolish taking the tin case stocked as it is, but experience has taught that even in the wildest and most out-of- the-way spots occasions arise when all pleasure is spoiled by not having the evening change of kit at hand. SECOND LIST: GROCERIES, ETC. Matches ½ doz. boxes of floats for oil lamp ½ doz. boxes of night-lights 6 lbs. of candles 8 to the lb. and 2 bedroom candlesticks 1 doz. tins of unsweetened tinned milk 1 lb. of tea ½ doz. coffee and milk in tins ½ doz. tins of chocolate and milk Plate powder Varnish for yellow leather shoes Corkscrew Sardine-box opener 3-lb. tin of marmalade Pepper Mustard
- 83. Jar of salt 1 doz. tins of sardines 3 tins of herring à la sardines 2 lb. captain's biscuits 1 doz. packages of jelly powder 6 doz. tinned soups Soap, 1 bar of common brown Soap, 1 bar of scented Wicks for stoves Plate, clothes, and boot brushes 2 chamois leathers. Cheese, butter, bread, ½ loaf per diem per man THIRD LIST: YACHT NECESSARIES Marlinespike Pricker Mop and twiddlers Hatchet Heavy hammer Small hammer Screw-driver Gimlet Bradawl Pincers Brass screws Copper nails Brass hooks 1 tin of black paint 1 tin of Harvey's Anti-fouling Paint 1 tin of Copal varnish Spare shackles, clip hooks, hooks and thimbles
- 84. 1 4-lb. lead and line 1 can of methylated spirits 1 can of mineral oil 1 can of colza oil Lamp showing red, white, or green, as required Riding light Binnacle and light 1 small-sized patent log 20 fathoms of Kaia grass warp 1 tail 4-in. block 2 spare blocks with hooks or thimbles 1 canvas bucket, medium size 1 iron bucket 2 brass holders for oil glass lamp, and to hold tumbler if required for flowers 4 thick common cups and saucers ½ doz. tumblers ½ doz. wineglasses 3 sodawater tumblers ½ doz. enamel plates ½ doz. enamel soup plates 2 enamel slop basins 2 enamel flat dishes 1 enamel double vegetable dish 1 deep dish for stews, c. 3 tablespoons 3 table forks 3 table knives ½ doz. small forks ½ doz. dessert spoons
- 85. ½ doz. teaspoons ½ doz. small knives Fish knife and fork 2 kitchen knives and forks 2 kitchen table- and 2 teaspoons Binocular glasses Parallel rulers Compasses Isle of Man almanac Charts: Irish Sea, West Scotland, English Channel, c. Books of sailing directions Channel pilot Flags: Club Burgee, Pilot Jack, and Ensign. The Pilot Jack is useful in case a pilot be required, and the Ensign to hoist upside down in case of distress, or in the rigging as a protest when racing Fishing tackle Medicine:— Brandy Friar's balsam Lint Bottle of Condy's fluid Carlsbad salts
- 86. Commercial Code of Signals. When used as the Code Signal this Pennant is to be hoisted under the Ensign. Such lists as are given above should be kept in a small book labelled 'Fitting-out Necessaries,' because they save much time at that season, and all alterations in them that experience dictates should be noted before or at the period of laying the yacht up. Racing. The yacht, let it be supposed, is fitted out. She has a racing outfit, and was the crack boat of the past season. There is a smart young fellow engaged to look after her, and the only thing that remains to be settled now is the question—Shall I give myself up to racing or shall I cruise this year? If it is to be racing, here are two or three
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