On the \(\phi_J\) polar decomposition of matrices
Define \(\phi_{s}:\mathbb{C}^{n\times n}\rightarrow\mathbb{C}^{n\times n}\) by \(\phi_{s}(A)=S^{-1}A^{T}S\), where \(S\in\mathbb{C}^{n\times n}\) is non-singular and skew symmetric. Then we say that \(A\in\mathbb{C}^{n\times n}\) has a \(\phi_{s}\) polar decomposition if \(A=QR\), where \(Q\) is \(\phi_{s}\) orthogonal (i.e., \(\phi_{s}(Q)=Q^{-1})\) and \(R\) is \(\phi_{s}\) symmetric (i.e., \(\phi_{s}(R)=R)\). If we define by \[ J:=\left( \begin{matrix} 0 & I_{n}\\ -I_{n} & 0 \end{matrix} \right) \in\mathbb{C}^{2n\times2n} \] then a \(\phi_{J}\)-orthogonal matrix is called \textit{symplectic}. Two matrices \(A,B\in\mathbb{C}^{2n\times2n}\) are called \textit{symplectic equivalent} if \(A=QBP\) for some symplectic matrices \(Q,P\in\mathbb{C}^{2n\times2n}\). This paper is a sequel of the authors' recent two papers [Linear Algebra Appl. 431, No. 8, 1249--1256 (2009; Zbl 1176.15016), Linear Algebra Appl. 430, No. 2--3, 756--761 (2009; Zbl 1157.15011)]. It starts by presenting that every symplectic matrix \(A\in\mathbb{C}^{2n\times2n}\) is a product of symplectic operation matrices. A simple form attained under symplectic equivalence, that makes easy to determine if a matrix does not have a \(\phi_{J}\) polar decomposition follows. Finally, the authors determine the matrices \(A\in\mathbb{C}^{2n\times2n}\) with rank 4 matrices that have a \(\phi_{J}\) polar decomposition.
scientific article; zbMATH DE number 5669173