The ψS polar decomposition

Let S∈Mn be nonsingular. We set for all nonsingular A∈Mn; a matrix A is called ψS symmetric if ψS(A)=A, it is called ψS orthogonal if ψS(A)=A-1, and it is called ψS antiorthogonal if ψS(A)=-A-1. We show that the following are equivalent: (1) A is ψS symmetric, (2) there exists a ψS antiorthogonal Z∈Mn such that A=eZ, (3) there exists a ψS orthogonal X∈Mn such that A=eiX, and (4) there exists a ψS symmetric B∈Mn such that A=B2 . When S is coninvolutory or skew-coninvolutory , we show that every nonsingular matrix has a ψS polar decomposition, that is, every nonsingular matrix may be written as A=RE, where R is ψS orthogonal and E is ψS symmetric. If A is possibly singular, we define A to be ψS orthogonal if and determine which singular matrices have a ψS polar decomposition.