Page 397 - Matrix Analysis & Applied Linear Algebra

5.9 Complementary Subspaces 393

in which the blocks are matrix representations of restricted operators as
shown below.
1 2 1 2
A 11 = PAP /X B X . A 12 = PA(I − P) /Y B Y B X .
1 2 1 2
A 21 = (I − P)AP /X B X B Y . A 22 = (I − P)A(I − P) /Y B Y .

n
5.9.16. Suppose that = X⊕ Y, where dim X = r, and let P be the
projector onto X along Y. Explain why there exist matrices X n×r
and A r×n such that P = XA, where rank (X)= rank (A)= r and
AX = I r . This is a full-rank factorization for P (recall Exercise 3.9.8).

5.9.17. For either a real or complex vector space, let E be the projector onto
X 1 along Y 1 , and let F be the projector onto X 2 along Y 2 . Prove
that E + F is a projector if and only if EF = FE = 0, and under this
condition, prove that R (E + F)= X 1 ⊕X 2 and N (E + F)= Y 1 ∩Y 2 .

5.9.18. For either a real or complex vector space, let E be the projector onto
X 1 along Y 1 , and let F be the projector onto X 2 along Y 2 . Prove
that E − F is a projector if and only if EF = FE = F, and under this
condition, prove that R (E − F)= X 1 ∩Y 2 and N (E − F)= Y 1 ⊕X 2 .
Hint: P is a projector if and only if I − P is a projector.

5.9.19. For either a real or complex vector space, let E be the projector onto
X 1 along Y 1 , and let F be the projector onto X 2 along Y 2 . Prove that
if EF = P = FE, then P is the projector onto X 1 ∩X 2 along Y 1 +Y 2 .

5.9.20. An inner pseudoinverse for A m×n is a matrix X n×m such that
AXA = A, and an outer pseudoinverse for A is a matrix X satis-
fying XAX = X. When X is both an inner and outer pseudoinverse,
X is called a reflexive pseudoinverse.
(a) If Ax = b is a consistent system of m equations in n un-
knowns, and if A − is any inner pseudoinverse for A, explain
why the set of all solutions to Ax = b can be expressed as
n
−
−
−
−
A b + R I − A A = {A b +(I − A A)h | h ∈ }.
(b) Let M and L be respective complements of R (A) and N (A)
m n
so that C = R (A) ⊕M and C = L⊕ N (A). Prove that
there is a unique reflexive pseudoinverse X for A such that
R (X)= L and N (X)= M. Show that X = QA P, where
−
A − is any inner pseudoinverse for A, P is the projector onto
R (A) along M, and Q is the projector onto L along N (A).