A four parameter extension to the Clement matrix and its role in numerical software testing

Abstract

The Sylvester–Kac matrix (also known as the Clement matrix) and its generalizations are of interest to researchers in diverse fields. A new parameterization of the matrix has recently been presented with closed forms for the eigenvalues and Oste and Van der Jeugt (2017) have proposed their family of matrices as a source of test problems for numerical eigensolvers. In this article we extend their generalization by adding a further two parameters to the matrix definition and, for this new extension, we obtain closed forms for the eigenvalues, determinant and, the left and right eigenvectors. We show that, for certain values of the free parameters and relatively low order, these matrices may become very ill-conditioned w.r.t. both eigenvalues and inversion. In light of this we re-assess the numerical results presented by Oste and Van der Jeugt (2017) and propose a possible new testing role for parameterized Clement matrices.