The phiS polar decomposition when the cosquare of S is nonderogatory

Abstract

For $S \in GL_n$, define $\phi_S: M_n \rightarrow M_n$ by $\phi_S(A) = S^{-1}A^TS$. A matrix $A \in M_n$ is $\phi_S$ \textit{orthogonal} if $\phi_S(A) = A^{-1}$; $A$ is $\phi_S$ \textit{symmetric} if $\phi_S(A) = A$; $A$ has a $\phi_S$ \textit{polar decomposition} if $A = ZY$ for some $\phi_S$ orthogonal $Z$ and $\phi_S$ symmetric $Y$. If $A$ has a $\phi_S$ polar decomposition, then $A$ commutes with the cosquare $S^{-T}S$. Conditions under which the converse implication holds for the case where $S^{-T}S$ is nonderogatory, are obtained.