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

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.

The interesting spectral interlacing property for a certain tridiagonal matrix

“…e.g. [5,21] and references therein). Recently, Coelho, Dimitrov, and Rakai in [8] suggested a method for a fast estimation of the largest eigenvalue of an asymmetric tridiagonal matrix.…”

“…Then they provided numerical results with simulations in C/C++ implementation in order The Interesting Spectral Interlacing Property for a Certain Tridiagonal Matrix to demonstrate the effectiveness of the proposed method. They adopted the Sylvester-Kac test matrix [21] for comparing the power method and the proposed method performance. We also refer to [24] for further usage of test matrices.…”

The spectrum of a new class of Sylvester-Kac matrices

“…, n. Since then, many extensions and proofs have been proposed. Perhaps the most pertinent results can be found in [2][3][4][7][8][9][10][11][12][13][14] and references therein. The matrix A n , which we call Sylvester-Kac matrix, became also known as Clement matrix due to the independent study of P.A.…”

“…Recently, R. Oste and J. Van der Jeugt [14] established a study of a new family of matrices. From this study we can deduce that the eigenvalues of…”

“…e.g. [1,14] and references therein). We believe that this family and the corresponding explicit eigenvalues will make a significant contribution to these types of special matrices.…”