The bounds of the eigenvalues for rank-one modification of Hermitian matrix

The eigenproblems of the rank-one updates of the matrices have lots of applications in scientific computation and engineering such as the symmetric tridiagonal eigenproblems by the divide-andconquer method and Web search engine. Many researchers have well studied the algorithms for computing eigenvalues of Hermitian matrices updated by a rank-one matrix [1–6]. Recently, Ding and Zhou studied a spectral perturbation theorem for rank-one updated matrices of special structure in [7] and considered two applications. Cheng, Luo, and Li considered the bounds of the smallest and largest eigenvalues for rank-one modification of Hermitian matrices [8]. Eigenvalue bounds for perturbations of Hermitian matrices have been considered by Ipsen and Nadler in [9]. In this paper, we consider the bounds of the eigenvalues for rank-one modification of Hermitian matrices. The ideas of this paper were mainly motivated by one of [9]. We study the following form