Nonnegative Matrices
Nonnegative Matrices
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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