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.