Matrix similarity transformations derived from extended $q$-analogues of the Toda equation and Lotka-Volterra system

TL;DR

This paper introduces extended $q$-Toda equations linked to matrix eigenvalue problems and demonstrates their application in time-discretized eigenvalue computations, also exploring similar extensions for the Lotka-Volterra system.

Contribution

It develops new $q$-analogues of the Toda and Lotka-Volterra systems and applies them to matrix eigenvalue problems and discretization methods.

Findings

Extended $q$-Toda equations relate to matrix eigenvalue problems

Time-discretization of these equations aids in eigenvalue computation

Analogous extensions are discussed for the Lotka-Volterra system

Abstract

The $q$-Toda equation is derived from replacing ordinary derivatives with $q$-derivatives in the famous Toda equation. In this paper, we associate an extension of the $q$-Toda equation with matrix eigenvalue problems, and then show applications of its time-discretization to computing matrix eigenvalues. With respect to the Lotka-Volterra system, we also have the similar discussion on the case of the Toda equation.