  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.
