A note on the eigenvalues of a Sylvester-Kac type matrix with off-diagonal biperiodic perturbations

Zhibin Du, Carlos M. da Fonseca

Abstract: The Sylvester–Kac matrix is a tridiagonal matrix with integer entries having a certain kind of regular pattern. Its eigenvalues and eigenvectors can be computed analytically, so it can be used for test matrices for eigenvalue solvers. Among the extensions of the Sylvester–Kac matrix, one of the most challenging is when a given constant is added biperiodically to the non-zero off-diagonal entries. In this note we provide a combinatorial proof for determining the eigenvalues of such matrices. We also discuss a possible biperiodic extension.