Exercise 7.D.12

Answers

Proof. As a counterexample, take the linear map $T$ from Exercise 5. We have

M ({(T^{2})}^{∗} T^{2}) = M {(T^{∗})}^{2} M {(T)}^{2} = (\begin{matrix} - 4 & 0 \\ 0 & - 4 \end{matrix}) (\begin{matrix} - 4 & 0 \\ 0 & - 4 \end{matrix}) = (\begin{matrix} 16 & 0 \\ 0 & 16 \end{matrix}) .

Therefore, the singular values of $T^{2}$ are $4, 4$ . However, the singular values of $T$ are $1, 4$ . □

celiopassos

2017-10-06 00:00