Tridiagonal test matrices for eigenvalue computations: two-parameter extensions of the Clement matrix

TL;DR

This paper introduces two-parameter extensions of the Clement matrix, providing explicit eigenvalues and demonstrating their effectiveness as test matrices for numerical eigenvalue computations.

Contribution

It presents new two-parameter tridiagonal matrices with explicit spectra, extending the classical Clement matrix for improved testing in eigenvalue algorithms.

Findings

Eigenvalues are explicitly given by simple formulas.

The new matrices serve as effective test cases for numerical eigenvalue methods.

Numerical experiments confirm their utility in eigenvalue computations.

Abstract

The Clement or Sylvester-Kac matrix is a tridiagonal matrix with zero diagonal and simple integer entries. Its spectrum is known explicitly and consists of integers which makes it a useful test matrix for numerical eigenvalue computations. We consider a new class of appealing two-parameter extensions of this matrix which have the same simple structure and whose eigenvalues are also given explicitly by a simple closed form expression. The aim of this paper is to present in an accessible form these new matrices and examine some numerical results regarding the use of these extensions as test matrices for numerical eigenvalue computations.