IMC 2002 · Problem 5

Let $A$ be an $n \times n$ matrix with complex entries and suppose that $n > 1$. Prove that

$$A\overline{A} = I_n \iff \exists S \in GL_n(\mathbb{C}) \quad \text{such that} \quad A = S\overline{S}^{-1}.$$

(If $A = [a_{ij}]$ then $\overline{A} = [\overline{a_{ij}}]$, where $\overline{a_{ij}}$ is the complex conjugate of $a_{ij}$; $GL_n(\mathbb{C})$ denotes the set of all $n \times n$ invertible matrices with complex entries, and $I_n$ is the identity matrix.)