Elementary symmetric functions of two solvents of a quadratic matrix equation

TL;DR

This paper introduces new symmetric functions of the two solutions of a quadratic matrix equation, linking them to difference equations and illustrating their application in a physical problem.

Contribution

It presents permutationally invariant functions of quadratic matrix equation solutions, extending classical symmetric functions to a matrix setting.

Findings

Defined new symmetric functions for matrix solutions

Established connection with noncommutative difference equations

Applied results to a physical problem

Abstract

Quadratic matrix equations occur in a variety of applications. In this paper we introduce new permutationally invariant functions of two solvents of the n quadratic matrix equation X^2- L1X - L0 = 0, playing the role of the two elementary symmetric functions of the two roots of a quadratic scalar equation. Our results rely on the connection existing between the QME and the theory of linear second order difference equations with noncommutative coefficients. An application of our results to a simple physical problem is briefly discussed.