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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.

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