why does singular values of M equal to square roots of eigenvalues $M^TM$

Question

per wiki, there is a rule to compute singular values

The non-zero singular values of M (found on the diagonal entries of Σ) are the square roots of the non-zero eigenvalues of both $M^*M$ and $MM^*$.

for simplicity, let M is an m*n matrix of real values.

why does this rule apply?

Answer 1

Let $M = U\Sigma V^{*}.$ Then what is $M^*M$? Given that $\Sigma$ is diagonal, what can you say about the eigenvalues of $M^*M$?

Hint: Eigenvalues are invariant under ...