Matrices 1.pdf

MATRICES
Dr. Gabriel Obed Fosu
Department of Mathematics
Kwame Nkrumah University of Science and Technology
Google Scholar: https://scholar.google.com/citations?user=ZJfCMyQAAAAJ&hl=en&oi=ao
ResearchGate ID: https://www.researchgate.net/profile/Gabriel_Fosu2
Dr. Gabby (KNUST-Maths) Matrices 1 / 35

Lecture Outline
1 Introduction
Matrix Arithmetic and Properties
2 Some Special Matrices
Nonsingular Matrices
Symmetric Matrices
Orthogonal and Orthonormal Matrix
3 Complex Matrices
Hermitian Matrices

Introduction Matrix Arithmetic and Properties
Definition
1 A matrix is a rectangular array of numbers, symbols, or anything with m rows and n columns
which is used to represent a mathematical object or a property of such an object. The symbol
Rm×n
denotes the collection of all m × n matrices whose entries are real numbers. Matrices
represent linear maps.
2 Matrices will usually be denoted by capital letters, and the notation A = [ai j ] specifies that the
matrix is composed of entries ai j located in the ith row and jth column of A. Example of 2×3
matrix
A =
·
1 2 −9
3 −3 2
¸
3 A vector is a matrix with either one row or one column.
Column vector (3×1):
x =


1
−3
7


Row vector (1×4):
y =
£
6 1 0 −12
¤

Square and Zero Matrices
Definition (Square matrix)
Matrices of size (n,n) are called square matrices or n-square matrices of order n. Examples of
2×2 and 3×3 matrices are respectively given as
·
1 2
3 4
¸
,


1 2 3
4 5 6
7 8 8


Definition (The zero matrix)
Each m ×n matrix, all of whose elements are zero, is called the zero matrix (of size m ×n) and is
denoted by the symbol 0.
S =
·
0 0
0 0
¸

Identity Matrix
Definition (The identity matrix)
The n × n matrix I = [δi j ], defined by δi j = 1 if i = j, and δi j = 0 if i ̸= j, is called the n × n identity
matrix of order n.
Example of 3 × 3 identity matrix is
I =


1 0 0
0 1 0
0 0 1


When any n ×n matrix A is multiplied by the identity matrix, either on the left or the right, the result
is A. Thus, the identity matrix acts like 1 in the real number system.

Definition (Equality of matrices)
Matrices A and B are said to be equal if they have the same size and their corresponding elements
are equal; i.e., A and B have dimension m × n, and A = [ai j ],B = [bi j ], with ai j = bi j for 1 ≤ i ≤
m, 1 ≤ j ≤ n.
Definition (Addition of matrices)
Let A = [ai j ] and B = [bi j ] be of the same size. Then A + B is the matrix obtained by adding
corresponding elements of A and B; that is,
A +B = [ai j ]+[bi j ] = [ai j +bi j ]
Definition (Scalar multiple of a matrix)
Let A = [ai j ] and t be a number (scalar). Then t A is the matrix obtained by multiplying all elements
of A by t; that is,
t A = t[ai j ] = [tai j ]

Definition (Negative of a matrix)
Let A = [ai j ]. Then −A is the matrix obtained by replacing the elements of A by their negatives.
Definition (Subtraction of matrices)
Matrix subtraction is defined for two matrices A = [ai j ] and B = [bi j ] of the same size, in the usual
way; that is,
A −B = [ai j ]−[bi j ]

Example
If
A =
·
1 2
3 4
¸
,B =
·
5 6
0 −1
¸
A +B =
·
6 8
3 3
¸
A −B =
·
−4 −4
3 5
¸
2B =
·
10 12
0 −2
¸

Properties
The matrix operations of addition, scalar multiplication, negation and subtraction satisfy the
following laws of arithmetic. Let s and t be arbitrary scalars and A,B,C be matrices of the same
size
1 (A +B)+C = A +(B +C)
2 A +B = B + A
3 0+ A = A
4 A +(−A) = 0
5 (s + t)A = sA + t A, (s − t)A = sA − t A
6 t(A +B) = t A + tB, t(A −B) = t A − tB
7 s(t A) = (st)A
8 1A = A, 0A = 0, (−1)A = −A
9 t A = 0 =⇒ t = 0 or A = 0

Matrix Product
Let A = [ai j ] be a matrix of size m × p and B = [bjk] be a matrix of size p ×n (i.e., the number of
columns of A equals the number of rows of B). Then the product AB is an m ×n matrix.
That is, if
A =
·
a b
c d
¸
,B =
·
e f
g h
¸
then
AB =
·
ae +bg a f +bh
ce +dg c f +dh
¸
Example
·
1 2
3 4
¸·
5 6
7 8
¸
=
·
19 22
43 50
¸
£
3 4
¤
·
1
2
¸
=
£
11
¤
Matrix multiplication is not commutative AB ̸= B A

Trace
Definition (Trace)
If A is an n × n matrix, the trace of A, written trace(A), is the sum of the main diagonal elements;
that is,
trace(A) = a11 + a22 +···+ ann =
n
X
i=1
aii
Example
If
A =
·
1 2
3 4
¸
,B =
·
5 6
0 −1
¸
then
trace(A) = 1+4 = 5
trace(B) = 5+(−1) = 4

Properties of Trace
trace(A +B) = trace(A)+ trace(B)
trace(c A) = c · trace(A), where c is a scalar.
trace(AB) = trace(B A)

Power of a Matrix
Definition (kth power of a matrix)
If A is an n ×n matrix, we define Ak
as follows:
A0
= I
and
Ak
= A × A × A··· A × A; A occurs k times for k ≥ 1.
Example
A4
= A × A × A × A

The transpose of a matrix
Definition (The transpose of a matrix)
Let A be an m ×n matrix. Then AT
, the transpose of A, is the matrix obtained by interchanging the
rows and columns of A. In other words if A = [ai j ], then (AT
)i j = aji . If
A =
·
1 2 3
0 −1 15
¸
then
AT
=


1 0
2 −1
3 15



Properties of Transpose
1 (AT
)T
= A
2 (A ±B)T
= AT
±BT
if A and B are m ×n
3 (sA)T
= sAT
if s is a scalar
4 (AB)T
= BT
AT
if A is m ×k and B is k ×n

Some Special Matrices
Definition (Diagonal Matrix)
The aii ,1 ≤ i ≤ n, entries of a square matrix are called the diagonal elements. If the nondiagonal
elements are all zero, then the matrix is called a diagonal matrix. It is denoted by A =
diag(a11,a22,...,ann). Some examples are
·
10 0
0 −13
¸
,


14 0 0
0 −9 0
0 0 3



Definition (Bidiagonal Matrix)
A bidiagonal matrix is a matrix with nonzero entries along the main diagonal and either the diagonal
above or the diagonal below.
The matrix B1 is an upper bidiagonal matrix.




5 1 0 0
0 10 −1 0
0 0 9 2
0 0 0 −2



 (1)
The matrix B2 is a lower bidiagonal matrix.




5 0 0 0
2 10 0 0
0 9 9 0
0 0 −1 −2



 (2)

Definition (Tridiagonal Matrix)
A tridiagonal matrix has only nonzero entries along the main diagonal and the diagonals above and
below. T is a 5×5 tridiagonal matrix.
A =






2 1 0 0 0
3 4 5 0 0
0 −1 −3 2 0
0 0 1 2 10
0 0 0 −6 7






(3)

Some Special Matrices Nonsingular Matrices
Nonsingular matrix
Definition (Nonsingular matrix)
An n ×n matrix A is called nonsingular or invertible if there exists an n ×n matrix B such that
AB = B A = I
The matrix B is the inverse of A. If A does not have an inverse, A is called singular.
Properties: Let denote the inverse of A by A−1
, then
1 (A−1
)A = I = A(A−1
)
2 (A−1
)−1
= A
3 (AB)−1
= B−1
A−1
4 if A is nonsingular, then AT
is also nonsingular and (AT
)−1
= (A−1
)T
Homogeneous system
A linear system Ax = 0 is said to be homogeneous. If A is nonsingular, then x = A−1
(0) = 0, so the
system has only 0 as its solution. It is said to have only the trivial solution.

Some Special Matrices Symmetric Matrices
Symmetric and Skew Symmetric matrix
Symmetric
A matrix A is symmetric if AT
= A.
Another way of looking at this is that when the rows and columns are interchanged, the resulting
matrix is A.
Skew symmetric
A matrix A is skew symmetric if AT
= −A.
Example
The matrix A =


2 −3 4
−3 1 2
4 2 3

 is symmetric and B =


0 2 1
−2 0 −3
−1 3 0

 is skew symmetric.

Symmetric Definite Matrices
Definition (Symmetric Positive Definite Matrix)
A symmetric matrix A is positive definite if for every nonzero vector x =





x1
x2
.
.
.
xn





xT
Ax > 0 (4)
The expression xT
Ax is called the quadratic form associated with A.
Note
The sum of two positive definite matrices is positive definite.

Definition (Symmetric Positive Semidefinite Matrix)
A is symmetric positive semidefinite if for every nonzero vector x =





x1
x2
.
.
.
xn





xT
Ax ≥ 0 (5)

Example
The symmetric matrix A =
·
1 0
0 1
¸
is positive definite because for
x =
·
x1
x2
¸
̸=
·
0
0
¸
(6)
then
xT
Ax = [x1 x2]
·
1 0
0 1
¸·
x1
x2
¸
(7)
= x2
1 + x2
2 (8)
Since x2
1 + x2
2 > 0 then A is a symmetric positive definite matrix.

Theorem
1 If A = (ai j ) is positive definite, then aii > 0 for all i.
2 If A = (ai j ) is positive definite, then the largest element in magnitude of all matrix entries must
lie on the diagonal.
Example
The matrix A =


1 2 3
4 0 1
2 5 6

 cannot be positive definite because A has a diagonal element of 0
Example
The matrix B =




1 −1 0 9
8 45 3 19
0 15 16 35
3 −55 2 22



 cannot be positive definite because the largest element in
magnitude (−55) is not on the diagonal of B.

Theorem
Suppose that a real symmetric tridiagonal matrix
A =









b1 a1
a1 b2 a2
a2
...
...
... bn−1 an−1
an−1 bn









(9)
with diagonal entries all positive is strictly diagonally dominant, that is,
bi > |ai−1|+|ai |, 1 ≤ i ≤ n
Then A is positive definite.

Definition (Symmetric Negative Definite Matrix)
A is symmetric negative definite if for every nonzero vector x =





x1
x2
.
.
.
xn





xT
Ax ≤ 0 (10)
In this case, −A is positive definite.
Definition (Symmetric Indefinite Matrix)
A is symmetric indefinite if xT
Ax assumes both positive and negative values.
Alternatively, a matrix is symmetric indefinite if it has both positive and negative eigenvalues.

Some Special Matrices Orthogonal and Orthonormal Matrix
Definition (Orthogonal Matrix)
A matrix P is orthogonal if
PT
P = I (11)
The inverse of P is its transpose.
Example
The matrix P =
·
cosθ −sinθ
sinθ cosθ
¸
is orthogonal because
PT
P =
·
cosθ sinθ
−sinθ cosθ
¸·
cosθ −sinθ
sinθ cosθ
¸
(12)
=
·
cos2
θ +sin2
θ 0
0 cos2
θ +sin2
θ
¸
(13)
=
·
1 0
0 1
¸
(14)

Theorem
The matrix P is orthogonal if and only if the columns of P are orthogonal and have unit length.
〈vi , vj 〉 =
(
1 if i = j
0 if i ̸= j
(15)
Example
D = [e1 e2 e3] =


1 0 0
0 1 0
0 0 1

 (16)
then
〈e1, e2〉 = 〈e1, e3〉 = 〈e2, e3〉 = 0 (17)
and
〈e1, e1〉 = 〈e2, e2〉 = 〈e3, e3〉 = 1 (18)
Hence D is an orthogonal matrix.

Definition (Orthonormal)
1 A set of orthogonal vectors, each with unit length, are said to be orthonormal.
2 D = [e1 e2 e3] is an orthogonal matrix, and each orthogonal vector e1, e2 and e2 has a unit length.
3 Hence e1, e2,··· ,en are called the standard orthonormal basis.
The 3×3 identity matrix is orthogonal, each column vectors has unit lenght, thus forms a set of
orthonormal basis
|e1| = |e2| = |e3| =
q
a2
11 + a2
21 + a2
31 =
p
12 = 1 (19)

Complex Matrices Hermitian Matrices
If a = 1+3i then the complex conjugate is ā = 1−3i
Definition (Complex conjugate matrix)
The complex conjugate of an n×m complex matrix A = (zi j ) is defined and denoted by Ā = (z̄i j )m×n.
Definition (Hermitian and Skew Hermitian matrix)
A complex n-square matrix A is said to be hermitian if
ĀT
= A or z̄ji = zi j (20)
and skew hermitian if
ĀT
= −A or z̄ji = −zi j (21)
Example (M is Hermitian and N is skew Hermitian)
M =


2 1−i 0
1+i −1 i
0 −i 2

 N =


i 2+i 3+2i
−2+i 3i −3i
−3+2i −3i 0



Complex Matrices Hermitian Matrices
Exercises
1 Let A, B, C, D be matrices defined by
A =


3 0
−1 2
1 1

,B =


1 5 2
−1 1 0
−4 1 3

,C =


−3 −1
2 1
4 3

,D =
·
4 −1
2 0
¸
Which of the following matrices are defined? Compute those matrices which are defined. A +
B, A +C, AB,B A,CD,DC,D2
,(CT
)T
2 Rotate the line y = −x +3 60°counterclockwise about the origin.
3 Let σ1 =
¡0 1
1 0
¢
, σ2 =
¡0 −i
i 0
¢
and σ1 =
¡1 0
0 −1
¢
be the three Pauli matrices. Show that xσ1 + yσ2 +2σ3
is a hermitian matrix for any two real numbers x, y ∈ R.

END OF LECTURE
THANK YOU

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