Perturbation Theory for Analytic Matrix Functions: The Semisimple Case
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Summary: The eigenvalue problem for non-self-adjoint, analytic matrix functions of two variables, \(L(\lambda,\alpha)\), is examined with emphasis on the case when, at a fixed \(\alpha_0\), \(L(\lambda,\alpha_0)\) has a multiple, semisimple eigenvalue \(\lambda_0\). New sufficient conditions for analytic dependence of eigenvalue functions, \(\lambda(\alpha)\), on \(\alpha\) in a neighborhood of \(\alpha_0\) are obtained. An algorithm for generating Taylor coefficients of perturbed eigenvalues and eigenvectors is studied and the existence of positive radii of convergence is established. Connections with known results on self-adjoint problems are made.
Numerical computation of eigenvalues and eigenvectors of matrices, Eigenvalues, singular values, and eigenvectors, algorithm, convergence, Perturbation theory of linear operators, Taylor coefficients, analytic matrix functions, eigenvectors, semisimple eigenvalues, Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), analytic dependence, Matrices over function rings in one or more variables, non-self-adjoint functions, perturbation theory
Numerical computation of eigenvalues and eigenvectors of matrices, Eigenvalues, singular values, and eigenvectors, algorithm, convergence, Perturbation theory of linear operators, Taylor coefficients, analytic matrix functions, eigenvectors, semisimple eigenvalues, Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), analytic dependence, Matrices over function rings in one or more variables, non-self-adjoint functions, perturbation theory
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