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A Schur decomposition for Hamiltonian matrices. (English) Zbl 1115.15316

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.

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices
15A24 Matrix equations and identities
65F05 Direct numerical methods for linear systems and matrix inversion
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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
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