Page 226 - Matrix Analysis & Applied Linear Algebra

4.5 More about Rank 221

4.5.17. The Frobenius Inequality. Establish the validity of Frobenius’s 1911
result that states that if ABC exists, then

rank (AB)+ rank (BC) ≤ rank (B)+ rank (ABC).

Hint: If M = R (BC)∩N (A) and N = R (B)∩N (A), then M⊆N.

4.5.18. If A is n × n, prove that the following statements are equivalent:
2
(a) N (A)= N A .
2
(b) R (A)= R A .
(c) R (A) ∩ N (A)= {0}.

2
2
4.5.19. Let A and B be n × n matrices such that A = A , B = B , and
AB = BA = 0.
(a) Prove that rank (A + B)= rank (A)+ rank (B). Hint: Con-
A
sider (A + B)(A | B).
B
(b) Prove that rank (A)+ rank (I − A)= n.

m×n
4.5.20. Moore–Penrose Inverse. For A ∈ such that rank (A)= r,
let A = BC be the full rank factorization of A in which B m×r is the
matrix of basic columns from A and C r×n is the matrix of nonzero
rows from E A (see Exercise 3.9.8). The matrix defined by

−1 T
T
T
T

A = C B AC B
†
30
is called the Moore–Penrose inverse of A. Some authors refer to
†
A as the pseudoinverse or the generalized inverse of A. A more elegant
treatment is given on p. 423, but it’s worthwhile to introduce the idea
here so that it can be used and viewed from different perspectives.
T
(a) Explain why the matrix B AC T is nonsingular.
T
T
(b) Verify that x = A b solves the normal equations A Ax = A b (as
†
well as Ax = b when it is consistent).
T
T
(c) Show that the general solution for A Ax = A b (as well as Ax = b
when it is consistent) can be described as

†
x = A b + I − A A h,
†
30
This is in honor of Eliakim H. Moore (1862–1932) and Roger Penrose (a famous contemporary
English mathematical physicist). Each formulated a concept of generalized matrix inversion—
Moore’s work was published in 1922, and Penrose’s work appeared in 1955. E. H. Moore is
considered by many to be America’s first great mathematician.