Skew-coninvolutory matrices

We study the properties of skew-coninvolutory matrices, and derive canonical forms and a singular value decomposition. We study the matrix function , defined on nonsingular matrices and with S satisfying or . We show that every square nonsingular A may be written as A=XY with ψS(X)=X and ψS(Y)=Y-1. We also give necessary and sufficient conditions on when a nonsingular matrix may be written as a product of a coninvolutory matrix and a skew-coninvolutory matrix or a product of two skew-coninvolutory matrices. Moreover, when A is similar to , or when A is similar to , or when A is similar to , or when A is similar to , we determine the possible Jordan canonical forms of A for which the similarity matrix may be taken to be skew-coninvolutory.