×

Normal forms for symplectic matrices. (English) Zbl 1304.15012

Normal forms for symplectic matrices are constructed using elementary geometrical methods. They are expressed in terms of elementary Jordan matrices and integers with values in \(\{- 1,0,1\}\) related to signatures of quadratic forms naturally associated to the symplectic matrix. They are useful to give new characterisations of Conley-Zehnder indices of general paths of symplectic matrices. The natural interpretation of the signs appearing in the decomposition and the description of the decomposition for matrices with 1 as an eigenvalue are of interest in this description.

MSC:

15A21 Canonical forms, reductions, classification
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Y.-H. Au-Yeung, C.-K. Li, and L. Rodman, H-unitary and Lorentz matrices: a re- view. SIAM J. Matrix Anal. Appl. 25 (2004), 1140-1162. · Zbl 1063.15021
[2] I. Gohberg, P. Lancaster, and L. Rodman, Indefinite linear algebra and applications. Birkhaüser Verlag, Basel 2005. · Zbl 1084.15005
[3] I. Gohberg and B. Reichstein, On H-unitary and block-Toeplitz H-normal operators. Linear and Multilinear Algebra 30 (1991), 17-48. · Zbl 0746.15016
[4] J. Gutt, Generalized Conley-Zehnder index. Ann. Fac. Sci. Toulouse Math. To appear. · Zbl 1330.37020
[5] A. J. Laub and K. Meyer, Canonical forms for symplectic and Hamiltonian matrices. Celestial Mech. 9 (1974), 213-238. · Zbl 0316.15005
[6] W.-W. Lin, V. Mehrmann, and H. Xu, Canonical forms for Hamiltonian and symplectic matrices and pencils. Linear Algebra Appl. 302/303 (1999), 469-533. · Zbl 0947.15004
[7] Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. Adv. Math. 154 (2000), 76-131. · Zbl 0970.37013
[8] Y. Long, Index theory for symplectic paths with applications. Progr. Math. 207, Birkhaüser Verlag, Basel 2002. · Zbl 1012.37012
[9] Y. Long and D. Dong, Normal forms of symplectic matrices. Acta Math. Sin. (Engl. Ser.) 16 (2000), 237-260. · Zbl 0959.15008
[10] C. Mehl, Essential decomposition of polynomially normal matrices in real indefinite inner product spaces. Electron. J. Linear Algebra 15 (2006), 84-106. · Zbl 1095.15011
[11] C. Mehl, On classification of normal matrices in indefinite inner product spaces. Electron. J. Linear Algebra 15 (2006), 50-83. · Zbl 1095.15010
[12] D. Mu\"ller and C. Thiele, Normal forms of involutive complex Hamiltonian matrices under the real symplectic group. J. Reine Angew. Math. 513 (1999), 97-114. · Zbl 0932.15006
[13] L. Rodman, Similarity vs unitary similarity and perturbation analysis of sign charac- teristics: complex and real indefinite inner products. Linear Algebra Appl. 416 (2006), 945-1009. · Zbl 1098.15023
[14] V. V. Sergeı\?chuk, Classification problems for systems of forms and linear mappings. Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 1170-1190; English transl. Math. USSR- Izv. 31 (1988), 481-501. · Zbl 0678.15011
[15] E. Spence, m-symplectic matrices. Trans. Amer. Math. Soc. 170 (1972), 447-457. · Zbl 0281.15007
[16] H. K. Wimmer, Normal forms of symplectic pencils and the discrete-time algebraic Riccati equation. Linear Algebra Appl. 147 (1991), 411-440. · Zbl 0722.15012
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.