probability problem with brief solution 3

STAT 714 HOMEWORK 3
1. Define the matrix
A =




1 1 0 0
1 1 0 0
0 0 1 0
0 0 1 1



 .
(a) Find two generalized inverses of A.
(b) Find a matrix which projects onto C(A).
(c) Find a matrix which projects onto C(A)⊥
, the orthogonal complement of C(A).
2. Show that if A−
is a generalized inverse of A, then so is
G = A−
AA−
+ (I − A−
A)B1 + B2(I − AA−
),
for any choices of B1 and B2 with conformable dimensions.
3. Let An×p, bp×1, cn×1, and suppose that the equations Ab = c are consistent. Let
xn×1, up×1, and Xp×n. Let A−
1 and A−
2 be two generalized inverses of A. Let I denote
the n × n identity matrix.
(a) Let b∗
be a solution to Ab = c. Show that b∗
+uc0
{(A−
1 )0
A0
−I}x is also a solution.
(b) Show that A−
1 + X(AA−
2 − I) is a generalized inverse of A.
4. Suppose the system Ax = c is consistent and that G is a generalized inverse of A.
(a) What is a particular solution to the system? the general solution?
(b) If A is symmetric, prove that 1
2
(G + G0
) is a generalized inverse of A.
(c) Prove that the generalized inverse in (b) is symmetric. This shows that there does
exist a generalized inverse of A, A symmetric, that is symmetric itself.
5. Suppose that A, B, and A + B are all idempotent. Prove that AB = 0 and BA = 0.
6. Let P be an n×n orthogonal matrix and let A be an n×n symmetric and idempotent
matrix. Define D = P0
AP. Show that D is a perpendicular projection matrix.
7. Consider the linear model Y = Xβ + with
Y =




1
−1
2
0



 and X =




1 1 0 0
1 0 1 0
1 0 1 0
0 0 1 1



 .
Note that r(X) = 3. Find b
β1 and b
β2, two different solutions to the normal equations
X0
Xβ = X0
Y. With your solutions, show that Xb
β1 = Xb
β2 ∈ C(X). Also show that
Y − Xb
β1 = Y − Xb
β2 ∈ N(X0
).
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STAT 714 HOMEWORK 3
8. Let M1 and M2 be perpendicular projection matrices on Rn
. Prove that M1 + M2 is
the perpendicular projection matrix onto C(M1, M2) if and only if C(M1)⊥C(M2).
9. Let M be the perpendicular projection matrix onto C(X). Suppose that a ∈ C(X).
Show that (M − aa0
)0
(M − aa0
) = M + (a0
a − 2)aa0
.
10. Suppose that M1 and M2 are symmetric, that C(M1)⊥C(M2), and that M1 + M2
is the perpendicular projection matrix. Prove that M1 and M2 are also perpendicular
projection matrices.
11. Let M and M0 be perpendicular projection matrices with C(M0) ⊂ C(M). Show
that M − M0 is a perpendicular projection matrix.
DEFINITION : Let V denote an arbitrary vector space and let S denote a subspace of
V. Define
S⊥
V = {y ∈ V : y⊥S}.
The subspace S⊥
V is called the orthogonal complement of S with respect to V. If
V = Rn
, then S⊥
V ≡ S⊥
; in this situation, we call S⊥
and S simply “orthogonal com-
plements” because it is understood that the larger vector space is Rn
. However, there is
nothing to prevent V from being a subspace of Rn
.
12. Let M and M0 be perpendicular projection matrices with C(M0) ⊂ C(M). Show
that C(M − M0) = C(M0)⊥
C(M), the orthogonal complement of C(M0) with respect to
C(M).
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probability problem with brief solution 3

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