On condiagonalizable matrices (Q886147)

Two matrices \(A,B\in M_{n}(\mathbb{C})\) are `consimilar' if there exists a nonsingular matrix \(S\) such that \(A=SB\overline{S}^{-1}\) [see \textit{R. A. Horn} and \textit{C. R. Johnson}, Matrix analysis, Cambridge University Press (1985; Zbl 0576.15001)]. The author defines \(A\) to be `condiagonalizable' if \(\bar{A}A\) is similar in the usual sense to a diagonal matrix. This differs from the definition given in the reference above. The author shows that if \(A\) is condiagonalizable then \(A\) is consimilar to a block diagonal matrix whose diagonal blocks are either real \(1\times1\) blocks or \(2\times2\) blocks of the forms \[ \left[ \begin{matrix} 0 & \bar{\mu}\\ \mu & 0 \end{matrix} \right] \quad\text{or}\quad \left[ \begin{matrix} 0 & 0\\ 1 & 0 \end{matrix} \right] . \]