I saw recently this weird definition in an exam:
A matrix $A\in\mathcal M_n(\Bbb C)$ is said to be co-diagonalizable if there exists an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PD\overline P^{-1}$.
The aim of the exam is to prove that
A matrix $A$ is co-diagonaizable if and only if $A\overline A$ is diagonalizable (in its classic sense), $\operatorname{sp}(A\overline A)\subset \Bbb R_{\ge0}$ and $\operatorname{rank}(A\overline A)=\operatorname{rank}(A)$.
Can somebody give me a reference (book, paper or web site) in which we can find this definition and the related result? Thanks