\(\phi_S\)-orthogonal matrices
For any \(S\in M_n(\mathbb{R})\) with \(S^2=I\) or \(S^2=-I\), define \(\phi_S(A)=S^{-1}A^TS\). A matrix \(A\in M_n(\mathbb{C})\) is called \(\phi_S\)-orthogonal if \(\phi_S(A)A=I\). In this paper, the authors show that any complex matrix \(A\) admits a polar decomposition \(A=PU\) such that \(P\) is positive definite, \(U\) is unitary and both factors are \(\phi_S\)-orthogonal. More detailed properties for \(\phi_S\)-orthogonal matrices are given when \(S\) is of the form \(H_k=\left(\begin{matrix} 0& I_k\\-I_k& 0\end{matrix}\right)\) or \(L_k=\left(\begin{matrix} I_k &0\\0& -I_{n-k}\end{matrix}\right)\).
scientific article; zbMATH DE number 5702825