ψS-orthogonal matrices and ψS-symmetries

Let GLn(C) denote the set of n-by-n nonsingular matrices with entries from the field C of complex numbers. For any S∈GLn(C), define the map ψS:GLn(C)→GLn(C) by ψS(A)=SA−1‾S−1. A matrix A∈GLn(C) is said to be ψS-orthogonal if ψS(A)=A−1. A ψS-orthogonal H is called a ψS-symmetry if rank(H−I)=1. We give conditions on S so that ψS-symmetries exist. Moreover, we determine conditions on S such that the ψS-symmetries generate the ψS-orthogonal matrices, and the minimum number of ψS-symmetries needed to express a ψS-orthogonal.