Isospectral Reductions of Non-negative Matrices

Isospectral reduction is an important tool for network/matrix analysis as it reduces the dimension of a matrix/network while preserving all its eigenvalues and eigenvectors. The main contribution of this manuscript is a proposed algorithmic scheme to approximate the stationary measure of a stochasti...

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Bibliographic Details
Published in:arXiv.org 2023-09
Main Authors: Baraviera, Alexandre, Duarte, Pedro, Longmei Shu, Torres, Maria Joana
Format: Article
Language:English
Subjects:
Eigenvalues
Eigenvectors
Iterative methods
Matrix methods
Reduction
Online Access:Get full text
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Summary:Isospectral reduction is an important tool for network/matrix analysis as it reduces the dimension of a matrix/network while preserving all its eigenvalues and eigenvectors. The main contribution of this manuscript is a proposed algorithmic scheme to approximate the stationary measure of a stochastic matrix based on isospectral reduction. This scheme can be advantageous when there is more than one eigenvalue near 1, precisely the case where iterative methods perform poorly. In addition we give a partial explanation why this scheme should work well, showing that in some situations isospectral reduction improves the spectral gap.
ISSN:2331-8422