Relationship between the characteristic polynomial and the spectrum of a diagonalizable matrix and those of its low-rank update

Abstract

Low-rank updated matrices are of crucial importance in many applications. Recently the relationship between the characteristic polynomial and the spectrum of a given matrix A and those of its specially structured rank-k updated matrix has become a hot topic. Many researchers consider the eigenproblem of a matrix of the form under the assumption that the columns of U _k or V _k are right or left eigenvectors corresponding to some non-defective eigenvalues of A. However, in many low-rank updated eigenproblems, this assumption does not hold. In this article, we investigate the low-rank updated eigenproblem without such a constraint; that is, our low-rank updates U _k , V _k  ∈ ℂ ^n×k can be any complex matrices such that is a rank-k matrix. We first consider the relationship between the characteristic polynomial of a diagonalizable matrix and that of its rank-k update. We then focus on two special cases of k = 1 and k = 2. Moreover, the spectral relationship between a diagonalizable matrix and its rank-1 and rank-2 updates is considered. Some applications of our results to the low-rank updated singular value problem are also discussed.

Keywords:

• low-rank update
• characteristic polynomial
• spectrum
• diagonalizable matrix
• eigenproblem

AMS Subject Classifications::

• 65F15
• 65F10

Acknowledgements

We would like to express our sincere thanks to Professor Fuzhen Zhang and the reviewers for providing us with insightful comments and invaluable suggestions that greatly improved the presentation of this article. The authors are grateful to Professors Moody Chu and Shu-fang Xu for their helpful discussions; and to Professor Jiu Ding for his preprints. Meanwhile, we thank Professor Xiao-qing Jin for his careful reading of this manuscript and many useful discussions. G. Wu is supported by the National Science Foundation of China under grants 10901132 and 11171289, the Qing-Lan Project of Jiangsu Province, and the 333 Project of Jiangsu Province. Y. Wei is supported by the National Natural Science Foundation of China under grant 10871051, Doctoral Program of the Ministry of Education under grant 20090071110003, 973 Program Project under Grant 2010CB327900, Shanghai Education Committee under Dawn Project 08SG01 and Shanghai Science & Technology Committee under grant 09DZ2272900.