Page 227 - Matrix Analysis & Applied Linear Algebra

222 Chapter 4 Vector Spaces

n×1
where h is a “free variable” vector in .

†
Hint: Verify AA A = A, and then show R I − A A = N (A).
†
−1
T T
†
(d) If rank (A)= n, explain why A = A A A .
(e) If A is square and nonsingular, explain why A = A −1 .
†
T
†
(f) Verify that A = C T B AC T −1 B T satisfies the Penrose equations:
T

AA A = A, AA † = AA ,
†
†
T
A AA = A , A A = A A.
†
†
†
†
†
Penrose originally defined A to be the unique solution to these four
†
equations.