Perturbation of Eigenvalues for Matrix Polynomials via The Bauer-Fike Theorems

In earlier papers, the Bauer--Fike technique [F. L. Bauer and C. T. Fike, Numer. Math., 2 (1960), pp. 137--144] was applied to the eigenvalue problem $A{\bf x} = \lambda {\bf x}$ [E. K.-W. Chu, Numer. Math., 49 (1986), pp. 685--691] and the generalized eigenvalue problem $A{\bf x} = \lambda B{\bf x}$ [E. K.-W. Chu, SIAM J. Numer. Anal., 24 (1987), pp. 1114--1125]. General multiple eigenvalues were dealt with and perturbation results were obtained for individual as well as clusters of eigenvalues. In this paper, we shall generalize the technique to the eigenvalue problem for matrix polynomials. Multiple eigenvalues for monic as well as regular matrix polynomials will be considered.

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