Quadratically normal and congruence-normal matrices

A matrix $A\in\mathbf C^{n\times n}$ is unitarily quasi-diagonalizable if $A$ can be brought by a unitary similarity transformation to a block diagonal form with $1\times1$ and $2\times2$ diagonal blocks. In particular, the square roots of normal matrices, the so-called quadratically normal matrices, are unitarily quasi-diagonalizable.
A matrix $A\in\mathbf C^{n\times n}$ is congruence-normal if $B=A\overline A$ is a conventional normal matrix. We show that every congruence-normal matrix $A$ can be brought by a unitary congruence transformation to a block diagonal form with $1\times1$ and $2\times2$ diagonal blocks. Our proof emphasizes and exploits the likeliness between the equations $X^2=B$ and $X\overline X=B$ for a normal matrix $B$. Bibl. – 13 titles.