Canonical forms for Hamiltonian and symplectic matrices and pencils

Linear Algebra and Its Applications

Volumn 302-303, Issue , 1999, Pages 469-533

  Lin, Wen Wei ^a   Mehrmann, Volker ^b   Xu, Hongguo ^c  

^a NATIONAL TSING HUA UNIVERSITY   (Taiwan)

^b CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

^c CASE WESTERN RESERVE UNIVERSITY   (United States)

Author keywords

Algebraic Riccati equation; Eigenvalue problem; Hamiltonian pencil (matrix); Jordan canonical form; Kronecker canonical form; Linear quadratic control; Symplectic pencil (matrix)

Indexed keywords

EID: 0000902795     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(99)00191-3     Document Type: Article

References (28)

1. P. Benner, V. Mehrmann, H.-G. Xu, A new method for computing the stable invariant subspace of a real Hamiltonian matrix, J. Comput. Appl. Math. 86 (1997) 17-43.
2. S. Bittanti, A.J. Laub, J.C. Willems (Eds.), The Riccati Equation, Springer, Heidelberg, 1991.
3. A. Bunse-Gerstner, Matrix factorization for symplectic methods, Linear Algebra Appl. 83 (1986) 49-77.
4. N. Burgoyne, R. Cushman, Normal forms for real Hamiltonian systems, in: C. Martin, R. Hermann (Eds.), The 1976 Ames Research Center Conference on Geometric Control Theory, Math Sci. Press, Brookline, Mass, 1977, pp. 483-529.
5. R. Byers, Hamiltonian and symplectic algorithms for the algebraic Riccati equation, Ph.D. Thesis, Cornell University, 1983.
6. K.-W. Chu, The solution of the matrix equation AXB - CXD = Y and (YA - DZ, YC - BZ) = (E, F), Linear Alg. Appl. 93 (1987) 93-105.
7. D.Z. Djokovic, J. Patera, P. Winternitz, H. Zassenhaus, Normal forms of elements of classical real and complex Lie and Jordan algebras, J. Math. Phys. 24 (1983) 1363-1374.
8. I. Gohberg, P. Lancaster, L. Rodman, Matrices and indefinite scalar products, Operator Theory, vol. 8, Birkhäuser Verlag, Basel, 1983.
9. A. Ferrante, B.C. Levy, Canonical forms for real symplectic matrix pencils, Linear Algebra Appl. 274 (1998) 259-300.
10. ∞ control theory, Lecture Notes in Control and Information Science, vol. 88, Springer, Heidelberg, 1987.
11. G. Freiling, Lecture Notes, 1997.
12. R. Gantmacher, Theory of Matrices, vols. I, II, Chelsea, New York, 1959.
13. G.H. Golub, C.F. Van Loan, Matrix Computations, third ed., The John Hopkins University Press, Baltimore, 1996.
14. R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
15. L. Kronecker, Algebraische Reduction der Schaaren bilinearer Formen, Sitzungsber. Akad. d. Wiss. Berlin, 1890, pp. 763-767.
16. P. Lancaster, M. Tismenetsky, The Theory of Matrices, second ed., Academic Press, Orlando, 1985.
17. P. Lancaster, L. Rodman, The Algebraic Riccati Equation, Oxford University Press, Oxford, 1995.
18. A.J. Laub, K. Meyer, Canonical forms for symplectic and Hamiltonian matrices, Celestial Mechanics 9 (1974) 213-238.
19. W.-W. Lin, T-C. Ho, On Schur type decompositions for Hamiltonian and symplectic pencils, Technical Report, Institute of Applied Mathematics, National Tsing Hua University, Taiwan, 1990.
20. V. Mehrmann, The autonomous linear quadratic control problem, Theory and Numerical Solution, Lecture Notes in Control and Information Science, vol. 163, Springer, Heidelberg, July 1991.
21. V. Mehrmann, A step towards a unified treatment of continous and discrete time control problems, Linear Algebra Appl. 241-243 (1996) 749-779.
22. J. Olson, H. Jensen, P. Jorgensen, Solution of large matrix equations which occur in response theory, J. Comp. Phys. 74 (1988) 65-282.
23. C.C. Paige, C.F. Van Loan, A Schur decomposition for Hamiltonian matrices, Linear Algebra Appl. 41 (1981) 11-32.
24. R.C. Thompson, The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil, Linear Algebra Appl. 14 (1976) 136-177.
25. R.C. Thompson, Pencils of complex and real symmetric and skew symmetric matrices, Linear Algebra Appl. 147 (1991) 323-371.
26. K. Weierstraß, Zur Theorie der bilinearen und quadratischen Formen, Monatsh. Akad. d. Wiss. Berlin, 1867, pp. 310-338.
27. H.K. Wimmer, The algebraic Riccati equation without complete controllability, SIAM J. Algebra Disc. Meth. 3 (1982) 1-12.
28. H.K. Wimmer, Normal forms of symplectic pencils and the discrete-time algebraic Riccati equation, Linear Algebra Appl. 147 (1991) 411-440.