An SVD-Like Matrix Decomposition and Its Applications

A matrix S ∈ C2m×2m is symplectic if SJS∗ = J , where J = [ 0 −Im Im 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 ∈ Rn×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 BJBT factorization of real skew-symmetric matrices. The BJBT factorization has the applications in solving the skew-symmetric systems of linear equations, and the eigenvalue problem for skew-symmetric/symmetric pencils. The BJBT 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.