Singular values of unitarily equivalent matrices

[Codezion]

Homework Statement

If two square matrices, A and B are unitarily equivalent then A = QBQ* for some unitary Q of the same size as A and B. Prove that A and B are unitarily equivalent if and only if they have the same singular values

Homework Equations

The Attempt at a Solution

I started from the definition of singular values:

Au = sigma v for singular vectors u and v
A*v = sigma u
substitution A with QBQ*,
QBQ*u = sigma v
(QBQ*)*v = QB*Q*v = sigma u

Can't see how this leads to proving that the sigmas of A and B are the same..?

 

[Dick]

If Q is unitary, the Q*=Q^(-1). Operate on the left by Q*.