On perturbations of matrix pencils with real spectra. II期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On perturbations of matrix pencils with real spectra. II

Authors:	Rajendra Bhatia Ren-Cang Li

Institution:	Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi -- 110016, India ; Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

Abstract:	A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following: Let 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.

Keywords:	Diagonalizable matrix pencil definite pencil real spectrum unitarily invariant norm perturbation bound

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