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K-Primitivity : A Literature Survey

Monimala Nej

Abstract

A nonnegative matrix A is said to be primitive if there exists a positive integer m such that entries in A^m are positive and smallest such m is called the exponent of A: Primitive matrices are useful in the study of finite Markov chains theory. In 1998, in the context of finite Markov chains, Ettore Fornasini and Maria Elena Valcher [6] extended the notion of primitivity for a nonnegative matrix pair (A;B) by considering a positive discrete homogeneous two-dimensional (2D) state model. Further generalization to this notion of primitivity for k-tuple (A1;A2;...;Ak) of nonnegative matrices A1;A2;...;Ak is quite natural and known as k-primitivity. In this paper we present various results on k-primitivity given by different researchers from time to time.

K-Primitivity : A Literature Survey

Abstract

A nonnegative matrix A is said to be primitive if there exists a positive integer m such that entries in A^m are positive and smallest such m is called the exponent of A: Primitive matrices are useful in the study of finite Markov chains theory. In 1998, in the context of finite Markov chains, Ettore Fornasini and Maria Elena Valcher [6] extended the notion of primitivity for a nonnegative matrix pair (A;B) by considering a positive discrete homogeneous two-dimensional (2D) state model. Further generalization to this notion of primitivity for k-tuple (A1;A2;...;Ak) of nonnegative matrices A1;A2;...;Ak is quite natural and known as k-primitivity. In this paper we present various results on k-primitivity given by different researchers from time to time.
Paper Structure (5 sections, 5 theorems, 5 equations, 2 figures)

This paper contains 5 sections, 5 theorems, 5 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Graph Theory
  3. Concept of $k$-primitivity and relevant results
  4. $K$-Primitivity of a digraph $D$
  5. Results on $2$-primitivity and $2$-exponent of $(A,B)$

Key Result

Proposition 1.1

Let $A$ be a positive matrix and $r=\rho(A).$ Then

Figures (2)

  • Figure 1: The star graph $K_{1,5}$ with a loop at the vertex $1$
  • Figure : $D(A)$ or $D(B).$

Theorems & Definitions (14)

  • Proposition 1.1: O. Perron Perr, 1907
  • Definition 1.1: H. Minc Minc
  • Example 1
  • Proposition 1.2: H. Minc Minc
  • Proposition 1.3: G. Frobenius Frob:2, 1912
  • Definition 1.2: see, H. Minc Minc
  • Theorem 1.4
  • Example 2
  • Proposition 2.1: V. Romanovsky Romanovskyyy
  • Example 3
  • ...and 4 more