The \(\phi_S\) polar decomposition
Let \(S_n^+ = \{S\in M_n(\mathbb{R})\mid S^2=1\}\), \(S_n^- = \{S\in M_n(\mathbb{R})\mid S^2=-1\}\), and \(S_n = S_n^+\cup S_n^-\). For \(S\in S_n\) let \(\phi_S : M_n(\mathbb{C})\rightarrow M_n(\mathbb{C})\); \(A\rightarrow S^{-1}A^TS\). With respect to \(\phi_S\), an element \(A\in M_n(\mathbb{C})\) is called symmetric if \(\phi_S (A) = A\), skew symmetric if \(\phi_S (A) = - A\), and orthogonal if \(A\) is nonsingular and \(\phi_S (A) = A^{-1}\). If \(A = QT\), where \(Q\) is orthogonal and \(T\) is symmetric, then \(A\) is said to have a \(\phi_S\) polar decomposition. The authors show that every nonsingular \(A\in M_n(\mathbb{C})\) such that \(\phi_S(\phi_S(A)) = A\) has a \(\phi_S\) polar decomposition. They also discuss possible Jordan canonical forms.
scientific article; zbMATH DE number 6127365