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Calculating the Eigenvectors of Diagonally Dominant Matrices

Calculating the Eigenvectors of Diagonally Dominant Matrices Calculating Dominant the Eigenvectors Matrices of Diagonally M. M. B L E V I N S AND G. W. STEWART The University of Texas at Austin, Austin, Texas ABSTRACT. An algorithm is proposed for calculating the eigenvectors of a diagonally dominant matrix all of whose elements are known to high relative accuracy. Eigenvectors corresponding to pathologically close eigenvalues are treated by computing the invariant subspace that they span. If the off-diagonal elements of the matrix are sufficiently small, the method is superior to standard techniques, and indeed it may produce a complete set of eigenvectors with an amount of work proportional to the square of the order of the matrix. An analysis is given of the effects of perturbations in the matrix on the eigenvectors. KEY WORDSAND PHRASES: eigenvalue, eigenvector, diagonal dominance, diagonally dominant CR CATEOORIES: 5.14 1. Introduction L e t A be a real m a t r i x of order n, and write A in the form A = D+E, where D = diag (81, ~2, " " , &) is the diagonal of A and E has zero diagonal elements. We shall say t h a t A is diagonally d o m http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the ACM (JACM) Association for Computing Machinery

Calculating the Eigenvectors of Diagonally Dominant Matrices

Journal of the ACM (JACM) , Volume 21 (2) – Apr 1, 1974

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References (19)

Publisher
Association for Computing Machinery
Copyright
Copyright © 1974 by ACM Inc.
ISSN
0004-5411
DOI
10.1145/321812.321821
Publisher site
See Article on Publisher Site

Abstract

Calculating Dominant the Eigenvectors Matrices of Diagonally M. M. B L E V I N S AND G. W. STEWART The University of Texas at Austin, Austin, Texas ABSTRACT. An algorithm is proposed for calculating the eigenvectors of a diagonally dominant matrix all of whose elements are known to high relative accuracy. Eigenvectors corresponding to pathologically close eigenvalues are treated by computing the invariant subspace that they span. If the off-diagonal elements of the matrix are sufficiently small, the method is superior to standard techniques, and indeed it may produce a complete set of eigenvectors with an amount of work proportional to the square of the order of the matrix. An analysis is given of the effects of perturbations in the matrix on the eigenvectors. KEY WORDSAND PHRASES: eigenvalue, eigenvector, diagonal dominance, diagonally dominant CR CATEOORIES: 5.14 1. Introduction L e t A be a real m a t r i x of order n, and write A in the form A = D+E, where D = diag (81, ~2, " " , &) is the diagonal of A and E has zero diagonal elements. We shall say t h a t A is diagonally d o m

Journal

Journal of the ACM (JACM)Association for Computing Machinery

Published: Apr 1, 1974

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