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Perturbations of Jordan matrices. (English) Zbl 1164.15004

The analysis of the eigenvalues of highly non-self-adjoint matrices and operators is now well developed. Such operators appear frequently in the study of certain PDEs, e.g. in fluid dynamics, and an understanding of their spectral behaviour, in particular under perturbations, is fundamental for stability studies. Important work in this direction has been done by V. B. Lidskii and others.
In this paper the authors go beyond Lidskii’s theory in two different directions. They consider perturbations of an \(N\times N\) Jordan block and focus on the asymptotic behaviour as \(N\to\infty\). In the first part of the paper the perturbation is assumed to be random and to have a perturbation parameter that is small but too large for the applicability of Lidskii’s theory. In the second part, the perturbation is assumed to have small rank but norm of order 1. In the case of random perturbations they can show that with increasing \(N\) most of the eigenvalues converge to a certain circle with centre at the origin. In the case of finite rank perturbations they completely determine the spectral asymptotics as \(N\) increases.

MSC:

15B52 Random matrices (algebraic aspects)
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[1] Dencker, N.; Sjöstrand, J.; Zworski, M., Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math., 57, 384-415 (2004) · Zbl 1054.35035
[2] Reddy, S. C.; Schmid, P. J.; Henningson, D. S., Pseudospectra of the Orr-Sommerfeld operator, SIAM J. Appl. Math., 53, 15-45 (1993) · Zbl 0778.34060
[3] Burke, J.; Overton, M., Stable perturbations of nonsymmetric matrices, Linear Algebra Appl., 171, 249-273 (1992) · Zbl 0756.15023
[4] Hager, M.; Sjöstrand, J., Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Ann., 342, 1, 177-243 (2008) · Zbl 1151.35063
[5] Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0148.12601
[6] Lidskii, V. B., Perturbation theory of non-conjugate operators, USSR Comput. Math. Math. Phys., 6, 73-85 (1966)
[7] Moro, J.; Burke, J. V.; Overton, M. L., On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure, SIAM J. Matrix Anal. Appl., 18, 793-817 (1997) · Zbl 0889.15016
[8] Ma, Y.; Edelman, A., Nongeneric eigenvalue perturbations of Jordan blocks, Linear Algebra Appl., 273, 45-63 (1998) · Zbl 0901.15006
[9] Trefethen, L. N.; Embree, M., Spectra and Pseudospectra: The Behaviour of Nonnormal Matrices and Operators (2005), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1085.15009
[10] Davies, E. B., Linear Operators and Their Spectra (2007), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1132.47002
[11] J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems, preprint 2003, http://www.math.berkeley.edu/ /ela.ps.gz; J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems, preprint 2003, http://www.math.berkeley.edu/ /ela.ps.gz
[12] Hager, M., Instabilité spectrale semiclassique d’operateurs non-autoadjoints II, Ann. Henri Poincaré, 7, 1035-1064 (2006) · Zbl 1115.81032
[13] Beitia, M. A.; Hoyos, I.; Zaballa, I., The change of the Jordan structure under one row perturbations, Linear Algebra Appl., 401, 119-134 (2005) · Zbl 1092.15007
[14] Edelman, A.; Ma, Y., Non-generic perturbations of Jordan blocks, Linear Algebra Appl., 273, 45-63 (1998) · Zbl 0901.15006
[15] Nevanlinna, R., Analytic Functions (1970), Springer-Verlag: Springer-Verlag Berlin, New York · JFM 62.0315.02
[16] E.B. Davies, P.A. Incani, Spectral properties of matrices associated with some directed graphs, preprint, Feb. 2008; E.B. Davies, P.A. Incani, Spectral properties of matrices associated with some directed graphs, preprint, Feb. 2008 · Zbl 1189.15011
[17] Simon, B., Rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle, J. Math. Anal. Appl., 329, 376-382 (2007) · Zbl 1110.33004
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