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Ans 6 -v1 - CONSTRUCTIVE PERTURBATION THEORY FOR MATRICES...

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arXiv:0905.4051v2 [math.SP] 26 May 2009 CONSTRUCTIVE PERTURBATION THEORY FOR MATRICES WITH DEGENERATE EIGENVALUES AARON WELTERS Abstract. Let A ( ε ) be an analytic square matrix and λ 0 an eigenvalue of A (0) of multiplicity m 1. Then under the generic condition, ∂ε det ( λI A ( ε )) | ( ε,λ )=(0 0 ) negationslash = 0, we prove that the Jordan normal form of A (0) corresponding to the eigenvalue λ 0 consists of a single m × m Jordan block, the perturbed eigenvalues near λ 0 and their eigenvectors can be represented by a single convergent Puiseux series containing only powers of ε 1 /m , and there are explicit recursive formulas to compute all the Puiseux series coefficients from just the derivatives of A ( ε ) at the origin. Using these recursive formulas we calculate the series coefficients up to the second order and list them for quick reference. This paper gives, under a generic condition, explicit recursive formulas to compute the perturbed eigenvalues and eigenvectors for non-selfadjoint analytic perturbations of matrices with degenerate eigenvalues. Key words. Matrix Perturbation Theory, Degenerate Eigenvalues, Perturbation of Eigenvalues and Eigenvectors, Puiseux Series, Recursive Formula, Characteristic Polynomial, Jordan Block AMS subject classification. 15A15, 15A18, 15A21, 41A58, 47A55, 47A56, 65F15, 65F40 1. Introduction. Consider an analytic square matrix A ( ε ) and its unperturbed matrix A (0) with a degenerate eigenvalue λ 0 . A fundamental problem in the analytic perturbation theory of non-selfadjoint matrices is the determination of the perturbed eigenvalues near λ 0 along with their corresponding eigenvectors of the matrix A ( ε ) near ε = 0. More specifically, let A ( ε ) be a matrix-valued function having a range in C n × n , the set of n × n matrices with complex entries, such that its matrix elements are analytic functions of ε in a neighborhood of the origin. Let λ 0 be an eigenvalue of the matrix A (0) with algebraic multiplicity m 1(which may be degenerate, i.e., m > 1). Then in this situation, it is well known [1, § 6.1.7], [2, § II.1.2] that for sufficiently small ε all the perturbed eigenvalues near λ 0 , called the λ 0 -group, and their corresponding eigenvectors may be represented as a collection of convergent Puiseux series, i.e., Taylor series in a fractional power of ε . What is not well known, however, is how we compute these Puiseux series when A ( ε ) is a non-selfadjoint analytic perturbation and λ 0 is a degenerate eigenvalue of A (0). There are sources on the subject like [4], [5], [6], [1, § 7.4], and [7, § 32], but it was found that there lacked explicit formulas, recursive or otherwise, to compute the series coefficients beyond the first order terms. Thus, the fundamental problem that this paper addresses is actually two-fold. First, find a method to determine how many Puiseux series there are that represent the λ 0 -group and their eigenvectors along with the fractional power of ε that is associated with each. And second, find explicit recursive formulas to compute all the series coefficients.
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