zbMATH — the first resource for mathematics

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).

15A18 Eigenvalues, singular values, and eigenvectors
15B48 Positive matrices and their generalizations; cones of matrices
Full Text: DOI
[1] Bapat, R.B.; Raghavan, T.E.S., Nonnegative matrices and applicaions, (1997), Cambridge University Press · Zbl 0879.15015
[2] Brin, S.; Page, L., The anatomy of a large-scale hypertextual web search engine, Comput. netw. ISDN syst., 30, 1-7, 107-117, (1998)
[3] L. Eldén, A note on the eigenvalues of the Google matrix, Report LiTH-MAT-R-04-01, 2003
[4] Horn, R.; Johnson, C.R., Cambridge university press, (1985)
[5] T.H. Haveliwala, S.D. Kamvar, The second eigenvalue of the Google matrix, Technical Report, Computer Science Department, Stanford University, 2003
[6] Kamvar, S.D.; Haveliwala, T.H.; Golub, G.H., Adaptive methods for the computation of pagerank, Linear algebra appl., 386, 51-65, (2004) · Zbl 1091.68044
[7] Langville, A.N.; Meyer, C.D., Deeper inside pagerank, Internet math., 1, 335-380, (2004) · Zbl 1098.68010
[8] Langville, A.N.; Meyer, C.D., A survey of eigenvector methods for web information retrieval, SIAM rev., 47, 135-161, (2005) · Zbl 1075.65053
[9] Minc, H., Nonnegative matrices, (1988), Wiley · Zbl 0638.15008
[10] L. Page, S. Brin, R. Motwani, T. Winograd, The PageRank citation ranking: Bringing order to the Web, Stanford Digital Library Working Papers, 1998
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.