Paige, Chris; Van Loan, Charles A Schur decomposition for Hamiltonian matrices. (English) Zbl 1115.15316 Linear Algebra Appl. 41, 11-32 (1981). Summary: A Schur-type decomposition for Hamiltonian matrices is given that relies on unitary symplectic similarity transformations. These transformations preserve the Hamiltonian structure and are numerically stable, making them ideal for analysis and computation. Using this decomposition and a special singular-value decomposition for unitary symplectic matrices, a canonical reduction of the algebraic Riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution. After presenting some real decompositions for real Hamiltonian matrices, we look into the possibility of an orthogonal symplectic version of the QR algorithm suitable for Hamiltonian matrices. A finite-step initial reduction to a Hessenberg-type canonical form is presented. However, no extension of the Francis implicit-shift technique was found, and reasons for the difficulty are given. Cited in 1 ReviewCited in 77 Documents MSC: 15B57 Hermitian, skew-Hermitian, and related matrices 15A24 Matrix equations and identities 65F05 Direct numerical methods for linear systems and matrix inversion Keywords:Schur-type decomposition; Hamiltonian matrices; singular-value decomposition; unitary symplectic matrices; algebraic Riccati equation PDFBibTeX XMLCite \textit{C. Paige} and \textit{C. Van Loan}, Linear Algebra Appl. 41, 11--32 (1981; Zbl 1115.15316) Full Text: DOI References: [1] Golub, G. H.; Reinsch, C. R., Singular value decomposition and least squares solutions, Numer. Math., 14, 403-420 (1970) · Zbl 0181.17602 [2] Lancaster, P.; Rodman, L., Existence and uniqueness theorems for the algebraic Riccati equation (Mar. 1979), Dept. of Mathematics and Statistics, Research Report No. 421, Univ. of Calgary [3] Laub, Alan J., A Schur method for solving algebraic Riccati equations, (Laboratory for Information and Decision Systems (Oct. 1978), M.I.T), Report LIDS-R-859 · Zbl 0427.65027 [4] Laub, Alan; Meyer, Kenneth, Canonical forms for symplectic and Hamiltonian matrices, J. Celestial Mechanics, 9, 213-238 (1974) · Zbl 0316.15005 [5] Moler, C. B.; Stewart, G. W., An algorithm for generalized eigenvalue problems, SIAM J. Numer. Anal., 10, 241-256 (1973) · Zbl 0253.65019 [6] Potter, J. E., Matrix quadratic solutions, SIAM J. Applied Math., 14, 496-501 (1966) · Zbl 0144.02001 [7] Stewart, G. W., Introduction to Matrix Computations (1973), Academic: Academic New York · Zbl 0302.65021 [8] Stewart, G. W., On the perturbation of pseudo-inverses, projections, and linear least squares problems, SIAM Rev., 19, 634-662 (1977) · Zbl 0379.65021 [9] Wilkinson, J. H., The Algebraic Eigenvalue Problem (1965), Oxford U.P: Oxford U.P London · Zbl 0258.65037 [10] Wonham, W. M., On a matrix Riccati equation of stochastic control, SIAM J. Control, 6, 681-697 (1968) · Zbl 0182.20803 [11] Minc, M.; Marcus, H., A Survey of Matrix Theory and Matrix Inequalities, ((1964), Prindle Weber and Schmidt: Prindle Weber and Schmidt Boston), 75 · Zbl 0126.02404 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.