On matrices whose exponential is a P-matrix

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.

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• Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull
• Generalization of the multiplicative and additive compounds of square matrices and contraction in the Hausdorff dimension