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Exercise 6.7.15

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1.

Use the fact that

A^{∗} A = P^{∗} W^{∗} WP = P^{2}

and

A A^{∗} = WP P^{∗} W^{∗} = W P^{2} W^{∗} .

So $A^{∗} A = A A^{∗}$ if and only if $P^{2} = W P^{2} W^{∗}$ , which is equivalent to $W P^{2} = P^{2} W$ .

2.

By the previous argument we have

A

is normal if and only if

P^{2} = W P^{2} W^{∗} = {(WP W^{∗})}^{2} .

Since $P$ and $WP W^{∗}$ are both positive semidifinite, again by Exercise 6.4.17(d), we have the condition is equivalent to $P = WP W^{∗}$ or $PW = WP$ .

jephianlin

2011-06-27 00:00

Exercise 6.7.15

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