Hi.

problem:Two matrices are unitarily equivalent if for some unitary .

Is it true or false that and are unitarily equivalent if and only if they have the same singular values?

attempt:

First direction:If , do and have the same singular values?

Let,

, and let,

.

Since , we have that;

.

This shows that has the same singular values as .

If and have the same singular values, are they unitarily equivalent?

Second direction:

Let,

, and let,

.

Then,

and so,

.

I would like to show that , but do not know how.

Interesting fact is that . Maybe I could use that in some way?

All suggestions are appreciated, thanks.