  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.

Numerical Analysis