singular values of product of normal matrices

Question

Let $A,B \in \mathbb{C}^{n\times n}$ be two normal matrices.

Is it true that:

$\sigma_{i}(AB) = \sigma_{i}(BA) \; \; \forall \; i$ where $\sigma_{i}$ is the $i$-th singular value of the corresponding matrix.

Any hint/comment/expalantion for proving this statement is appreciated.

Answer 1

Hints.

1. The singular values of a square matrix $X$ are the square roots of the eigenvalues of $X^\ast X$.
2. For any two complex square matrices $M,N$ of the same sizes, $MN$ and $NM$ have identical spectra.
3. Normal matrices commute with their complex conjugates.