An SVD-like matrix decomposition and its applications

A matrix S is an element of C-2m x 2m is symplectic if S J S* = 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 is an element of R-n x 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(Omega, Omega(-1)) and Omega 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. (C) 2003 Elsevier Science Inc. All fights reserved.