An SVD-like matrix decomposition and its applications

Abstract

A matrix S∈ C 2m×2m is symplectic if SJS ∗=J , where J= 0 I m −I m 0 . Symplectic matrices play an important role in the analysis and numerical solution of matrix problems involving the indefinite inner product x ∗( iJ)y . In this paper we provide several matrix factorizations related to symplectic matrices. We introduce a singular value-like decomposition B= QDS −1 for any real matrix B∈ R n×2m , where Q is real orthogonal, S is real symplectic, and D is permuted diagonal. We show the relation between this decomposition and the canonical form of real skew-symmetric matrices and a class of Hamiltonian matrices. We also show that if S is symplectic it has the structured singular value decomposition S=UDV ∗ , where U, V are unitary and symplectic, D= diag(Ω,Ω −1) and Ω is positive diagonal. We study the BJB T factorization of real skew-symmetric matrices. The BJB T factorization has the applications in solving the skew-symmetric systems of linear equations, and the eigenvalue problem for skew-symmetric/symmetric pencils. The BJB T factorization is not unique, and in numerical application one requires the factor B with small norm and condition number to improve the numerical stability. By employing the singular value-like decomposition and the singular value decomposition of symplectic matrices we give the general formula for B with minimal norm and condition number.