On matrix perturbations with minimal leading Jordan structure.
type: Item
instance of: scholarly article
title: On matrix perturbations with minimal leading Jordan structure.
zbMATH Open document ID: 1049.15004
author: Claude-Pierre Jeannerod
published in: Journal of Computational and Applied Mathematics
publication date: 2004-01-26T00:00:00Z
review text: Let \(A(\varepsilon )=A_0+A_1\varepsilon +O(\varepsilon ^2)\) be a matrix perturbation of an \(n\times n\) nilpotent matrix \(A_0\) over the set of complex numbers. Assuming that the Jordan canonical form of the leading matrix \(A_0\) is known, \textit{V. B. Lidskij} [Perturbation theory of non-conjugate operators. U.S.S.R. Comput. Math. Math. Phys., 6, 73--85 (1966); transl. from Zh. Vychisl. Mat. Mat. Fiz. 6, No. 1, 52--60 (1966; Zbl 0166.40501)] provided sufficient conditions on some entries of \(A_1\) for \(A(\varepsilon )\) to admit \(rm\) eigenvalues with order \(O(\epsilon ^{1/m})\) when \(A_0\) has \(r\) nilpotent Jordan blocks with size \(m.\) However Lidskij's conditions are not necessary and the same eigenvalue splitting may occur for some other leading Jordan structures. The author proves that any matrix perturbation of an \(n\times n\) nilpotent complex matrix is similar to a matrix perturbation whose leading coefficient has minimal Jordan structure. In addition, the author derive the property that, for matrix perturbations with minimal leading Jordan structure, the sufficient conditions of Lidskij's theorem are also necessary. The main tools for deriving these results are the Newton diagram associated with the characteristic polynomial of a matrix perturbation and the Newton envelope associated with its leading matrix.
reviewed by: Jaspal Singh Aujla
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 2032616
zbMATH Keywords: matrix perturbations
zbMATH Keywords: nilpotent Jordan structure
zbMATH Keywords: eigenvalues
zbMATH Keywords: Newton diagram
zbMATH Keywords: matrix similarity
zbMATH Keywords: Jordan canonical form
MaRDI profile type: Publication
full work available at URL: j.cam.2003.08.021
OpenAlex ID: W1966603336
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DOI: 10.1016/J.CAM.2003.08.021
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description: scientific article; zbMATH DE number 2032616