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Linear Algebra

Exercise 6.7.18

Answers

1.

Observe that if

A = U Σ V^{∗}

is a singular value decomposition of

A

, then

GA = (GU) Σ V^{∗}

is a single value decomposition of

GA

. So we have

{(GA)}^{†} = V Σ^{†} {(GU)}^{∗} = A^{†} G^{∗} .

2.

Observe that if

A = U Σ V^{∗}

is a singular value decomposition of

A

, then

AH = U Σ V^{∗} H^{∗∗} = U Σ {(H^{∗} V)}^{∗}

is a single value decomposition of

AH

. So we have

{(AH)}^{†} = H^{∗} V Σ^{†} U^{∗} = H^{∗} A^{†} .

jephianlin

2011-06-27 00:00

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