The ϕS polar decomposition

Let Sn+=S∈MnR:S2=I,Sn-=S∈MnR:S2=-I, and let Sn=Sn+∪Sn-. For S∈Sn, let ϕS:MnC→MnC be given by ϕSA=S-1ATS. An A∈MnC is called ϕSsymmetric if ϕSA=A;A is called ϕSskew symmetric if ϕSA=-A; and A is called ϕSorthogonal if A is nonsingular and ϕSA=A-1. We say that A has a ϕSpolar decomposition if A=QT for some ϕS orthogonal Q∈MnC and some ϕS symmetric T∈MnC. Let VS=A∈MnC:ϕSϕSA=A. We show that every nonsingular A∈VS has a ϕS polar decomposition. We determine conditions for which an A∈MnC has a ϕS polar decomposition. We also determine the possible Jordan Canonical Forms of a ϕS orthogonal matrix and of a ϕS skew symmetric matrix.