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

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.