The ψS polar decomposition when the cosquare of S is normal

Let a nonsingular S∈Mn(C)S∈Mn(C) be given. For a nonsingular A∈Mn(C)A∈Mn(C), set ψS(A)=S−1A‾−1S. We say that an A   is ψSψSorthogonal   if ψS(A)=A−1ψS(A)=A−1 and we say that A   is ψSψSsymmetric   if ψS(A)=AψS(A)=A. For a possibly singular B∈Mn(C)B∈Mn(C), we say that B   is ψSψSorthogonal   if S−1B‾S=B; we say that B   has a ψSψSpolar decomposition   if B=REB=RE for some (possibly singular) ψSψS orthogonal R   and (necessarily nonsingular) ψSψS symmetric E  . If S=IS=I, then the ψSψS polar decomposition is the real-coninvolutory decomposition. We show that if A is nonsingular, then A   has a ψSψS polar decomposition if and only if A   commutes with S‾S. Because S is nonsingular, the cosquare of S   (that is, S−TSS−TS) is normal if and only if S‾S is normal [11, Theorem 5.2]. In this case, we show that a possibly singular A∈Mn(C)A∈Mn(C) has a ψSψS polar decomposition if and only if (a) rank(A)rank(A) and rank((S‾S−λI)A) have the same parity for every negative eigenvalue λ   of S‾S, and (b) the ranges of SA   and A‾ are the same.