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Eigenvalues of rank-one updated matrices with some applications. (English) Zbl 1139.15003
Authors’ abstract: We prove a spectral perturbation theorem for rank-one updated matrices of special structure. Two applications of the result are given to illustrate the usefulness of the theorem. One is for the spectrum of the Google matrix and the other is for the algebraic simplicity of the maximal eigenvalue of a positive matrix. The main idea behind the proof is from a simple relation between the determinants of a matrix and its rank one perturbation (cf. Lemma 1.1 in the paper).

MSC:
15A18 Eigenvalues, singular values, and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
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