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Hidden Markov ProcessesTheory and Applications to Biology$
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M. Vidyasagar

Print publication date: 2014

Print ISBN-13: 9780691133157

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691133157.001.0001

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Nonnegative Matrices

Nonnegative Matrices

Chapter:
(p.71) Chapter Three Nonnegative Matrices
Source:
Hidden Markov Processes
Author(s):

M. Vidyasagar

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691133157.003.0003

This chapter deals with nonnegative matrices, which are relevant in the study of Markov processes because the state transition matrix of such a process is a special kind of nonnegative matrix, known as a stochastic matrix. However, it turns out that practically all of the useful properties of a stochastic matrix also hold for the more general class of nonnegative matrices. Hence it is desirable to present the theory in the more general setting, and then specialize to Markov processes. The chapter first considers the canonical form for nonnegative matrices, including irreducible matrices and periodic irreducible matrices, before discussing the Perron–Frobenius theorem for primitive matrices and for irreducible matrices.

Keywords:   nonnegative matrices, Markov process, canonical form, irreducible matrices, periodic irreducible matrices, Perron–Frobenius theorem, primitive matrices

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