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

An n × n complex matrix A is called coninvolutory if A ¯ A = I n and skew-coninvolutory if A ¯ A = − I n (which implies that n is even). We prove that each matrix of size n × n with n > 1 is a sum of 5 coninvolutory matrices and each matrix of size 2 m × 2 m 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 = S ¯ − 1 D S in which S is nonsingular and D is diagonal. [ABSTRACT FROM AUTHOR]