On matrices whose exponential is a P-matrix

TL;DR

This paper introduces the class of matrices whose exponential is always a P-matrix, exploring their properties and applications in opinion dynamics, thus linking matrix exponential behavior with P-matrix characteristics.

Contribution

The paper defines and analyzes the new class of matrices in~$ ext{EP}$, where the exponential is a P-matrix for all positive times, and applies this to opinion dynamics.

Findings

Matrices in~$ ext{EP}$ have specific structural properties.

The exponential of matrices in~$ ext{EP}$ remains a P-matrix over time.

Application to opinion dynamics demonstrates practical relevance.

Abstract

A matrix is called a P-matrix if all its principal minors are positive. P-matrices have found important applications in functional analysis, mathematical programming, and dynamical systems theory. We introduce a new class of real matrices denoted~$\EP$. A matrix is in~$\EP$ if and only if its matrix exponential is a P-matrix for all positive times. In other words, $A\in \EP$ if and only if the transition matrix of the linear system~$\dot x=Ax$ is a P-matrix for any positive time~$t$. We analyze the properties of this new class of matrices and describe an application of our theoretical results to opinion dynamics.