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Mathematics of Computation

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On perturbations of matrix pencils
with real spectra. II

Authors: Rajendra Bhatia and Ren-Cang Li
Journal: Math. Comp. 65 (1996), 637-645
MSC (1991): Primary 15A22, 15A42, 65F15
MathSciNet review: 1333304
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Abstract: A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following: Let $A$ and $% \widetilde A$ be two $n\times n$ Hermitian matrices, and let $% \lambda_1,\ldots,\lambda_{n}$ and $% \widetilde\lambda_1,\ldots,\widetilde\lambda_{n}$ be their eigenvalues arranged in ascending order. Then $% \left|\kern-1.5truept\left|\kern-1.5truept\left| \operatorname{diag}\, (\lambda_1-% \widetilde\lambda_1,\ldots,\lambda_n-% \widetilde\lambda_n)% \right|\kern-1.5truept\right|\kern-1.5truept\right| \le% \left|\kern-1.5truept\left|\kern-1.5truept\left| A-\widetilde A% \right|\kern-1.5truept\right|\kern-1.5truept\right|$ for any unitarily invariant norm $% \hbox{$|\kern-1.5truept|\kern-1.5truept|\cdot |\kern-1.5truept|\kern-1.5truept|$}$. In this paper, we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra. The much studied case of definite pencils is included in this.

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Additional Information

Rajendra Bhatia
Affiliation: Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi – 110016, India

Ren-Cang Li
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

Keywords: Diagonalizable matrix pencil, definite pencil, real spectrum, unitarily invariant norm, perturbation bound
Received by editor(s): April 13, 1994
Received by editor(s) in revised form: August 4, 1994
Additional Notes: The first author thanks NSERC Canada for financial support.
Article copyright: © Copyright 1996 American Mathematical Society