Singular-value-like decomposition for complex matrix triples

The classical singular value decomposition for a matrix A∈Cm×nA∈Cm×n is a canonical form for AA that also displays the eigenvalues of the Hermitian matrices AA∗AA∗ and A∗AA∗A. In this paper, we develop a corresponding decomposition for AA that provides the Jordan canonical forms for the complex symmetric matrices AAT and ATA. More generally, we consider the matrix triple (A,G,Gˆ), where G∈Cm×m,Gˆ∈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,Gˆ)↦(XTAY,XTGX,YTGˆY), where X,YX,Y are nonsingular.