APPENDIX B
THE PERRON-FROBENIUS THEOREM AND STOCHASTIC MATRICES
B.1 Introduction
A matrix possessing the property that all of its components are nonnegative is
referred to as a nonnegative matrix. For the matrix T, this condition is indicated by T 2 0.
The case in which all components of T are strictly positive is indicated by T > 0. This
notation extends in the obvious manner to expressions such as T 2 B < T- B > 0 relating
nonnegative matrices with compatible dimensions.
The definitions, theorems and corollary in Section B.2 below concern nonnegative
matrices. They are the foundation for the Markov chain stationary distribution existence
and representation theorem and related results summarized in Appendix A and employed
in Sections 2, 4, 7 and 8. They are extracted from [Sene81], and are specialized in Section
B.3 from the case of finite nonnegative matrices to the case of finite stochastic matrices.
They are employed extensively in Sections 6 and 7.
B.2 The Perron-Frobenius Theorem and Ancillary Results for Primitive Matrices
Theorem B2 below is called the strong version of the Perron-Frobenius theorem. It
applies to a class of nonnegative matrices referred to as primitive. A version of the theo-
rem which applies to the wider class of irreducible nonnegative matrices is usually
invoked for applications involving stochastic matrices, but the flexibility of the more
general version is not required for the purposes herein. The connection of these results to
those of Appendix A is provided by Theorem B1. It asserts that primitivity (Definition
Bl) is equivalent to the combination of irreducibility and aperiodicity, as defined in
Appendix A.