Originally Posted by

**Mollier** Ok, first of all I see that they have the same singular values.

I'll try this first:

$\displaystyle \begin{bmatrix}1&1\\0&1\end{bmatrix}=Q \begin{bmatrix}1&0\\0&1\end{bmatrix}Q^{*}=QQ^{*}=I

$

So that does not work.

$\displaystyle \begin{bmatrix}1&0\\0&1\end{bmatrix}=Q \begin{bmatrix}1&1\\0&1\end{bmatrix}Q^{*}

$

does not work either, because then say, $\displaystyle \begin{bmatrix}1&1\\0&1\end{bmatrix}Q^{*}

$ would have to be the inverse of $\displaystyle Q$, which can not be the case.

So the answer to the "second direction" is no. Even though a counter example is great, is there a way to come to that conclusion without using an example?

Perhaps by adding something to the work I have already done?