Singular-value-like decomposition for complex matrix triples
Please always quote using this URN:urn:nbn:de:0296-matheon-4224
- The classical singular value decomposition for a matrix $A\in\Cmn$ is a canonical form for $A$ that also displays the eigenvalues of the Hermitian matrices $AA^\ast$ and $A^\ast A$. In this paper, we develop a corresponding decomposition for $A$ that provides the Jordan canonical forms for the complex symmetric matrices $AA^T$ and $A^TA$. More generally, we consider the matrix triple $(A,G_1,G_2)$, where $G_1\in\CC{m}, G_2\in\CC{n}$ are invertible and either complex symmetric and complex skew-symmetric, and we provide a canonical form under transformations of the form $(A,G_1,G_2)\mapsto(X^T A Y, X^T G_1X, Y^T G_2Y)$, where $X,Y$ are nonsingular.