Bounds for eigenvalues of matrix polynomials
نویسندگان
چکیده
Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices. They generalize various existing bounds for scalar polynomials and single matrices. A variety of tools are used in the derivations, including block companion matrices, Gershgorin’s theorem, the numerical radius, and associated scalar polynomials. Numerical experiments show that the bounds can be surprisingly sharp on practical problems. © 2002 Elsevier Science Inc. All rights reserved. AMS classification: 65F15; 15A18
منابع مشابه
Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs
Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signles...
متن کاملSharp Bounds on the PI Spectral Radius
In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.
متن کاملAPPLICATION OF THE RANDOM MATRIX THEORY ON THE CROSS-CORRELATION OF STOCK PRICES
The analysis of cross-correlations is extensively applied for understanding of interconnections in stock markets. Variety of methods are used in order to search stock cross-correlations including the Random Matrix Theory (RMT), the Principal Component Analysis (PCA) and the Hierachical Structures. In this work, we analyze cross-crrelations between price fluctuations of 20 company stocks...
متن کاملComputing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method
We study the Lanczos method for computing extreme eigenvalues of a symmetric or Hermitian matrix. It is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues—even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we compute probabilistic bounds for the extreme eigenvalues from data available dur...
متن کاملComputing Characteristic Polynomials from Eigenvalues
This paper concerns the computation of the coefficients ck of the characteristic polynomial of a real or complex matrixA. We analyze the forward error in the coefficients ck when they are computed from the eigenvalues of A, as is done by MATLAB’s poly function. In particular, we derive absolute and relative perturbation bounds for elementary symmetric functions, which we use in turn to derive p...
متن کامل