Page 432 - Matrix Analysis & Applied Linear Algebra

428 Chapter 5 Norms, Inner Products, and Orthogonality

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5.12.16. Establish the following properties of A .
(a) A = A −1 when A is nonsingular.
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(b) (A ) = A.
† T T †
(c) (A ) =(A ) .
T −1 T
(A A) A when rank (A m×n )= n,
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(d) A = T T −1
A (AA ) when rank (A m×n )= m.
T
T
(e) A = A AA = A AA T for all A ∈ m×n .
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T
T
(f) A = A (AA ) =(A A) A T for all A ∈ m×n .
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T †
T
(g) R A † = R A = R A A , and
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T
N A † = N A = N AA .
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T
(h) (PAQ) = Q A P T when P and Q are orthogonal matrices,
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but in general (AB) = B A (the reverse-order law fails).
T
T †
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T †
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T †
(i) (A A) = A (A ) and (AA ) =(A ) A .
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5.12.17. Explain why A = A D if and only if A is an RPN matrix.
m×n
5.12.18. Let X, Y ∈ be such that R (X) ⊥ R (Y).
(a) Establish the Pythagorean theorem for matrices by proving
2
2
2
X + Y = X + Y .
F F F
(b) Give an example to show that the result of part (a) does not
hold for the matrix 2-norm.
(c) Demonstrate that A is the best approximate inverse for A
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in the sense that A is the matrix of smallest Frobenius norm
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that minimizes I − AX .
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