Singular-value-like decomposition for complex matrix triples

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