dc.contributor.author | Xu, Hongguo | |
dc.date.accessioned | 2005-05-02T15:11:19Z | |
dc.date.available | 2005-05-02T15:11:19Z | |
dc.date.issued | 2003-07-15 | |
dc.identifier.citation | Xu, HG. An SVD-like matrix decomposition and its applications. LINEAR ALGEBRA AND ITS APPLICATIONS. July 15 2003. 368:1-24. | |
dc.identifier.other | ISI:000183665000001 | |
dc.identifier.uri | http://hdl.handle.net/1808/374 | |
dc.description.abstract | 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. | |
dc.description.sponsorship | This author is supported by NSF under Grant No.EPS-9874732 and matching support from the State of Kansas. | |
dc.format.extent | 237991 bytes | |
dc.format.mimetype | application/pdf | |
dc.language.iso | en_US | |
dc.publisher | ELSEVIER SCIENCE INC | |
dc.subject | Skew-symmetric matrix | |
dc.subject | Symplectic matrix | |
dc.subject | Orthogonal (unitary) symplectic matrix | |
dc.subject | Hamiltonian matrix | |
dc.subject | Eigenvalue problem | |
dc.subject | Singular value decomposition (svd) | |
dc.subject | Svd-like decomposition | |
dc.subject | Bjb(t) factorization | |
dc.subject | Schur form | |
dc.subject | Jordan canonical form | |
dc.title | An SVD-like matrix decomposition and its applications | |
dc.type | Article | |
dc.identifier.doi | 10.1016/S0024-3795(03)00370-7 | |
dc.rights.accessrights | openAccess | |