Page 396 - Matrix Analysis & Applied Linear Algebra

392 Chapter 5 Norms, Inner Products, and Orthogonality

5.9.12. Let P and Q be projectors.
(a) Prove R (P)= R (Q)if and only if PQ = Q and QP = P.
(b) Prove N (P)= N (Q)if and only if PQ = P and QP = Q.
(c) Prove that if E 1 , E 2 ,..., E k are projectors with the same range,

and if α 1 ,α 2 ,...,α k are scalars such that j α j =1, then

α j E j is a projector.
j
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5.9.13. Prove that rank (P)= trace (P) for every projector P defined on .
Hint: Recall Example 3.6.5 (p. 110).

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5.9.14. Let {X i } i=1 be a collection of subspaces from a vector space V, and
let B i denote a basis for X i . Prove that the following statements are
equivalent.
(i) V = X 1 + X 2 + ··· + X k and X j ∩ (X 1 + ··· + X j−1 )= 0 for
each j =2, 3,...,k.
(ii) For each vector v ∈V, there is one and only one way to write
v = x 1 + x 2 + ··· + x k , where x i ∈X i .
(iii) B = B 1 ∪B 2 ∪· · ·∪B k with B i ∩B j = φ for i = j is a basis
for V.
Whenever any one of the above statements is true, V is said to be the
direct sum of the X i ’s, and we write V = X 1 ⊕X 2 ⊕· · ·⊕X k . Notice
that for k =2, (i) and (5.9.1) say the same thing, and (ii) and (iii)
reduce to (5.9.3) and (5.9.4), respectively.

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5.9.15. For complementary subspaces X and Y of , let P be the projec-
tor onto X along Y, and let Q =[X | Y]in which the columns of
X and Y constitute bases for X and Y, respectively. Prove that if

Q −1 A n×n Q is partitioned as Q −1 AQ = A 11 A 12 , then
A 21 A 22

A 11 0 −1 0A 12 −1
Q Q =PAP, Q Q = PA(I − P),
0 0 0 0

0 0 −1 0 0 −1
Q Q =(I − P)AP, Q Q =(I − P)A(I − P).
A 21 0 0A 12
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This means that if A is considered as a linear operator on , and if
B = B X ∪B Y , where B X and B Y are the respective bases for X and
Y defined by the columns of X and Y, then, in the context of §4.8, the

matrix representation of A with respect to B is [A] B = A 11 A 12
A 21 A 22