Exercise 6.7.14

Answers

Consider

A^{2} = A^{∗} A = V Σ U^{∗} U Σ V^{∗} = V Σ^{2} V^{∗} = {(V Σ V^{∗})}^{2} .

Both of $A$ and $V Σ V^{∗}$ are positive definite, we know that $A = V Σ V^{∗}$ by Exercise 6.4.17(d). So we know

V Σ V^{∗} = U Σ V^{∗} .

Since $A$ is positive definite, we know $Σ$ is invertible. Also, $V$ is invertible. Hence we get $U = V$ finally.

jephianlin

2011-06-27 00:00