Singular-value-like decomposition for complex matrix triples
Instead of computing the singular values of a general matrix \(A \in \mathbb C^{m \times n}\) and thus revealing the eigenvalues of \(AA^*\) and \(A^*A\), one may ask for a canonical form for \(A\) that reveals the eigenvalues of the complex symmetric matrices \(AA^T\) and \(A^TA\). The authors obtain such a form by solving a more general problem considering a triple of matrices \((A,G, \hat G)\) with \(A \in \mathbb C^{m \times n}, G \in \mathbb C^{m \times m}\) and \(\hat G \in \mathbb C^{n \times n}\), where \(G\) and \(\hat G\) are nonsingular and either complex symmetric or complex skew-symmetric. Then they derive canonical forms for matrix triples \((A, G, \hat G)\) that provide the Jordan canonical forms of the structured matrices \(\hat {\mathcal H} = \hat G^{-1}A^TG^{-1}A\) and \({\mathcal H} = G^{-1}A\hat G^{-1}A^T\).
scientific article; zbMATH DE number 5647946