ϕS-Orthogonal matrices

AbstractLet S∈Mn(R) be such that S2=I or S2=-I. For A∈Mn(C), define ϕS(A)=S-1ATS. We study ϕS-orthogonal matrices (those A∈Mn(C) that satisfy ϕS(A)=A-1). Let F=R or F=C. We show that every ϕS-orthogonal A∈Mn(F) has a polar decomposition A=PU with P,U∈Mn(F),P is positive definite, U is unitary, and both factors are ϕS-orthogonal. We show that if A is ϕS-orthogonal and normal, and if −1 is not an eigenvalue of A, then there exists a normal ϕS-skew symmetric N (that is, ϕS(N)=-N) such that A=eiN. We also take a look at the particular cases S=Hk≡0Ik-Ik0 and S=Lk≡Ik⊕-In-k