Geometric Interpretation and Manifold Structure of Markov Matrices

Abstract

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Andreyevich Markov. Every Markov matrix gives a linear equation system, and the solution of this equation system gives us a subset of $\mathbb{R}_{n}^{n}$. This paper presents the new manifold structure on the set of the Markov matrices. In addition, this paper presents the set of Markov matrices is drawable, and this gives geometrical interpretation to Markov matrices. For the proof, we use the one-to-one corresponding among $n \times n$ Markov matrices, the solution of linear equation system from derived Markov property, and the set of $(n-1)$-polytopes.