On a decomposition of partitioned \(J\)-unitary matrices (Q2915739)

The author gives a decomposition of a hyperbolic block-unitary matrix. His main result is the following theorem.NEWLINENEWLINELet \(U= \left[\begin{matrix} U_{11} & U_{12} \\ U_{21} & U_{22} \end{matrix}\right],\,\, J=J_1 \bigoplus J_2,\,\, J_1, J_2 = \text{diag}(\pm1),\) where \(U_{11}\) and \(J_1\) are square matrices of the same order, as well as \(U_{22}\) and \(J_2\).NEWLINENEWLINEIf \(U\) is a \(J\)-unitary matrix such that \(U_{11} \in \mathbb{C}^{n_1\times n_1}\) and \(U_{22} \in \mathbb{C}^{n_2\times n_2}\) have the \(2HSVD\) with regard to \(J_1\) and \(J_2\), respectively, it can be written in the formNEWLINENEWLINENEWLINE\[NEWLINEU=R\Delta, \,\,R=\left[\begin{matrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{matrix}\right],\,\, \Delta=\left[\begin{matrix} \Delta_1 & \\ & \Delta_2 \end{matrix}\right],NEWLINE\]NEWLINE where \(\Delta_1\) and \(\Delta_2\) are \(J_1\)- and \(J_2\)-unitary (respectively) and NEWLINE\[NEWLINER_{11}= \sqrt{I+R_{12}R_{21}},\,\, R_{21}=-J_2R_{12}^*J_1,\,\,R_{22}= \sqrt{I+R_{21}R_{12}}. NEWLINE\]NEWLINENEWLINENEWLINEBelow we explain some of the terms used in the theorem.NEWLINENEWLINEThe two-sided hyperbolic singular value decomposition (\(2HSVD\)) decomposes a given matrix \(A\) into \(A=U \Sigma V^{[\ast]}\), where \(U\) and \(V\) are \(J\)-unitary and \(\Sigma\) is real diagonal. Here \(V^{[\ast]}= JV^{\ast}J\) is the \(J\)-adjoint of \(V\).NEWLINENEWLINEIf \(X\) is a matrix with no negative eigenvalues, such that zero is their at most non-defective eigenvalue, then NEWLINE\[NEWLINE\sqrt{X}=\sum_{k=0}^{\infty} \left( \begin{matrix} 1/2 \\ k \end{matrix} \right)(X-I)^k.NEWLINE\]