Embeddable Markov Matrices

TL;DR

This paper explores properties of embeddable Markov matrices, including eigenvalue constraints and optimal approximation methods, providing both historical context and new results in the theory of Markov semigroups.

Contribution

It presents new insights into the eigenvalue regions for embeddable matrices and proves the optimality of a known approximation procedure.

Findings

Eigenvalues of embeddable matrices lie in a specific region within the unit ball.

A known approximation method for non-embeddable matrices is proven to be optimal.

The paper consolidates old and new results on Markov matrix embeddability.

Abstract

We give an account of some results, both old and new, about any $n\times n$ Markov matrix that is embeddable in a one-parameter Markov semigroup. These include the fact that its eigenvalues must lie in a certain region in the unit ball. We prove that a well-known procedure for approximating a non-embeddable Markov matrix by an embeddable one is optimal in a certain sense.