Each n-by-n matrix with n>1 is a sum of 5 coninvolutory matrices

Abstract

An $n\times n$ complex matrix $A$ is called coninvolutory if $\bar AA=I_n$ and skew-coninvolutory if $\bar AA=-I_n$ (which implies that $n$ is even). We prove that each matrix of size $n\times n$ with $n>1$ is a sum of 5 coninvolutory matrices and each matrix of size $2m\times 2m$ is a sum of 5 skew-coninvolutory matrices. We also prove that each square complex matrix is a sum of a coninvolutory matrix and a condiagonalizable matrix. A matrix $M$ is called condiagonalizable if $M=\bar S^{-1}DS$ in which $S$ is nonsingular and $D$ is diagonal.