Is the singular values of a positive semi-definite matrix the same as the multiplication of it by an arbitrary unitary matrix?

Question

I was studying my quantum information textbook and I wanted to solve an exercise, I had to prove something and in that proof, I used two facts that I thought is true, but today my TA said that it is not true. So I was wondering about that. There were two statements,

1. Consider $\sigma$ is positive semi-definite and $M$ is hermitian. I had used polar decomposition as follows $\sigma^{1/2}M\sigma^{1/2}=UP$ which U is unitary and $P$ is positive semi-definite. So we have $P=U^*\sigma^{1/2}M\sigma^{1/2}$. Now, can we say that $Tr(U^*\sigma^{1/2}M\sigma^{1/2})=Tr(\sigma^{1/2}M\sigma^{1/2})$? if yes, how? And is there any reference for proof of that?
2. Assume that $\Sigma_X$ is a diagonal matrix containing the singular values of X in order from largest to smallest, And assume again that we have a unitary matrix like U and a positive semi-definite like $\sigma$. So now the question is that is it true that we say $\Sigma_{U^*\sigma^{1/2}}=\Sigma_{\sigma^{1/2}}$? if yes, how? And is there any reference for proof of that?

Answer 1

Statement 1 is not correct, but statement 2 is correct. To see that 2 is correct, it suffices to note that the singular values of a matrix $X$ are the square roots of the eigenvalues of $X^*X$, and $$ [U^*\sigma^{1/2}]^*[U^*\sigma^{1/2}] = \sigma^{1/2} U^{**}U^* \sigma^{1/2} = \sigma = [\sigma^{1/2}]^* [\sigma^{1/2}]. $$