Page 165 - Matrices theory and applications

8. Matrix Factorizations
148
is positive (semi)definite if and only if S −W is, where S is the Schur
complement of A in H.
7. (continuation of exercise 6)
Fix the size k and denote by S(H) the Schur complement in the
Hermitian matrix H when A ∈ HPD n−k . Using the previous exercise,
show that:
(a) S(H + H ) − S(H) − S(H ) is positive semidefinite.

(b) If H − H is positive semidefinite, then so is S(H) − S(H ).

In other words, H → S is “concave nondecreasing” on the convex
set formed of those matrices of H n such that A ∈ HPD n−k into the
ordered set H k . The article [26] gives a review of the properties of
the map H → S(H).
8. In Proposition 8.3.1, find an alternative proof of the uniqueness part,
by inspection of the spectrum of the matrix Q := Q −1 Q 1 = R 2 R −1 .
2 1
9. Identify the generalized inverse of row matrices and column matrices.
10. What is the generalized inverse of an orthogonal projector, that is, a
2
Hermitian matrix P satisfying P = P? Deduce that the description
†
†
of AA and A A as orthogonal projectors does not characterize A †
uniquely.
p
11. Given a matrix B ∈ M p×q (CC) and a vector a ∈ CC ,let us form the
matrix A := (B, a) ∈ M p×(q+1) (CC).
†
(a) Let us define d := B a, c := a − Bd,and

c , if c =0,
†
b :=
2 −1 ∗
†
(1 + |d| ) d B , if c =0.
Prove that

B − db
†
†
A = .
b
2
(b) Deduce an algorithm (Greville’s algorithm in O(pq )operations
for the computation of the generalized inverse of a p × q matrix.
Hint: To get started with the algorithm, use Exercise 9.