SEEMOUS 2021 Problem 3

Let $A \in \mathcal{M}_n(\mathbb{C})$ be a matrix such that $(AA^*)^2=A^*A$ , where $A^*=(\bar{A})^t$ denotes the Hermitian transpose (i.e., the conjugate transpose) of $A$ . (a) Prove that $AA^*=A^*A$ . (b) Show that the non-zero eigenvalues of $A$ have modulus one.