Transformation to versal deformations of matrices

Linear Algebra and Its Applications

Volumn 337, Issue 1-3, 2001, Pages 87-108

  Mailybaev, Alexei A ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

Author keywords

Jordan algebra; Lie algebra; Normal form; Reversible matrix; Transformation; Versal deformation

Indexed keywords

EID: 0035501549     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(01)00346-9     Document Type: Article

References (31)

1. Arnold V.I. On matrices depending on parameters. Russian Math. Surveys. 26:1971;29-43.
2. Arnold V.I. Lectures on bifurcations and versal families. Russian Math. Surveys. 27:1972;54-123.
3. Arnold V.I. Geometrical Methods in the Theory of Ordinary Differential Equations. 1983;Springer, New York.
4. Sevryuk M.B. Arnold's Problems. 2000 (in Russian);PHASIS, Moscow.
5. Beitia M.A., Gracia J.M. Sylvester matrix equation for matrix pencils. Linear Algebra Appl. 232:1996;155-197.
6. Berg J.M., Kwatny H.G. A canonical parametrization of the Kronecker form of a matrix pencil. Automatica. 31:1995;669-680.
7. Burke J.V., Overton M.L. Stable perturbations of nonsymmetric matrices. Linear Algebra Appl. 171:1992;249-273.
8. Chevalley C. Theory of Lie Groups. I. 1946;Princeton University Press, Princeton.
9. 4 J. Differential Equations. 64:1986;340-374.
10. Djoković D.Z., Patera J., Winternitz P., Zassenhaus H. Normal forms of elements of classical real and complex Lie and Jordan algebras. J. Math. Phys. 24:1983;1363-1374.
11. Edelman A., Elmroth E., Kågström B. A geometric approach to perturbation theory of matrices and matrix pencils. Part I: Versal deformations. SIAM J. Matrix Anal. Appl. 18:1997;653-692.
12. Galin D.M. On real matrices depending on parameters. Uspekhi Mat. Nauk. 27:1972;241-242. in Russian.
13. Galin D.M. Versal deformations of linear Hamiltonian systems. Amer. Math. Soc. Transl. 118:1982;1-12.
14. F.R. Gantmacher, The Theory of Matrices, vol. I, Chelsea, New York, 1959.
15. García-Planas M.I., Magret D. Deformation and stability of triples of matrices. Linear Algebra Appl. 254:1997;159-192.
16. García-Planas M.I., Sergeichuk V.V. Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils. Linear Algebra Appl. 302-303:1999;45-61.
17. Mailybaev A.A. On tangent cones to the stability domain of a family of real matrices. Moscow Univ. Mech. Bulletin. 53:1998;20-22.
18. Mailybaev A.A. Transformation of families of matrices to normal forms and its application to stability theory. SIAM J. Matrix Anal. Appl. 21:2000;396-417.
19. Mailybaev A.A., Seiranyan A.P. The stability domains of Hamiltonian systems. J. Appl. Math. Mech. 63:1999;545-555.
20. Mailybaev A.A., Seyranian A.P. On singularities of a boundary of the stability domain. SIAM J. Matrix Anal. Appl. 21:2000;106-128.
21. Patera J., Rousseau C. Complex orthogonal and symplectic matrices depending on parameters. J. Math. Phys. 23:1982;705-714.
22. Patera J., Rousseau C. Versal deformations of elements of classical Jordan algebras. J. Math. Phys. 24:1983;1375-1380.
23. Patera J., Rousseau C., Schlomiuk D. Versal deformations of elements of real classical Lie algebras. J. Phys. A: Math. Gen. 15:1982;1063-1086.
24. Postnikov M. Lie Groups and Lie Algebras. 1986;Mir, Moscow.
25. D.S. Schmidt, Transformations to versal normal form, in: K.R. Meyer, D.S. Schmidt (Eds.), Computer Aided Proofs in Analysis, IMA Vol. Math. Appl., vol. 28, Springer, New York, 1989, pp. 235-240.
26. 4 J. Comput. Appl. Math. 52:1994;155-176.
27. Sevryuk M.B. Reversible linear systems and their versal deformations. J. Soviet Math. 60:1992;1663-1680.
28. Stolovitch L. On the computation of a versal family of matrices. Numer. Algorithms. 4:1993;25-46.
29. Thompson J.M.T. Instabilities and Catastrophes in Science and Engineering. 1982;Wiley, New York.
30. Wilkinson J.H. The Algebraic Eigenvalue Problem. 1965;Clarendon Press, Oxford.
31. Yakubovich V.A., Starzhinskii V.M. Parametric Resonance in Linear Systems. 1987;Nauka, Moscow. in Russian.