Singular-value-like decomposition for complex matrix triples

The classical singular value decomposition for a matrix A ∈ Cm×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA∗ and A∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices AA and AA. More generally, we consider the matrix triple (A,G, Ĝ), where G ∈ Cm×m, Ĝ ∈ Cn×n are invertible and either complex symmetric or complex skew-symmetric, and we provide a canonical form under transformations of the form (A,G, Ĝ) 7→ (XAY,XGX,Y T ĜY ), where X,Y are nonsingular.