Note on the equations of diffusion operators associated to a positive matrix

TL;DR

This paper develops a general framework for diffusion operators linked to positive matrices, analyzing their properties and applications to improve convergence in matrix iteration schemes.

Contribution

It introduces a novel approach to decompose matrix-vector products at the entry level, enhancing understanding and convergence analysis of diffusion operators.

Findings

Describes a general framework for diffusion operators

Establishes properties of state vectors in diffusion processes

Improves convergence proofs for fixed point problems

Abstract

In this paper, we describe the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We show how this can be applied to prove and improve the convergence of a fixed point problem associated to the matrix iteration scheme. The approach can be understood as a decomposition of the matrix-vector product operation in elementary operation at the vector entry level.