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Numerical Methods for Structured Markov Chains$
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Dario A. Bini, Guy Latouche, and Beatrice Meini

Print publication date: 2005

Print ISBN-13: 9780198527688

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198527688.001.0001

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MATRIX EQUATIONS AND CANONICAL FACTORIZATIONS

MATRIX EQUATIONS AND CANONICAL FACTORIZATIONS

Chapter:
(p.45) 3 MATRIX EQUATIONS AND CANONICAL FACTORIZATIONS
Source:
Numerical Methods for Structured Markov Chains
Author(s):

D. A. Bini

G. Latouche

B. Meini

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198527688.003.0003

This chapter is concerned with infinite block Toeplitz matrices, their relationships with matrix power series and matrix Laurent power series, and the fundamental problem of solving matrix equations and computing canonical factorizations. It introduces the concept of a Wiener algebra and provides a natural theoretical framework where the convergence properties of algorithms for solving Markov chains can easily be proved. Furthermore, the notion of canonical factorization provides a powerful tool for solving infinite linear systems, such as the fundamental system involving the stationary probability distribution of structured Markov chains.

Keywords:   block Toeplitz matrices, matrix power series, matrix Laurent power series, matrix equations, canonical factorizations, Wiener algebra, infinite linear systems

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