The open question of the relation between square matrix’s eigenvalues and its similarity matrix’s singular values in linear discrete system

Abstract

In linear discrete system, we often need to utilize similarity transformation to extend the solution ranges of its corresponding discrete matrix equations. Consequently, how to choose the similarity transformation matrix remains an open question in recent years. In this paper, by applying the theory of matrix’s Jordan canonical form and the related properties of nonnegative matrices, we construct the similarity transformation matrix of some special similarity transformation, then present a necessary and sufficient condition and a corresponding algorithm, thus solve the open question totally.