Sylvester–Kac matrices with quadratic spectra: A comprehensive note

Sylvester–Kac matrices are tridiagonal integral matrices with integral spectra or with eigenvalues presenting some kind of regularity. Recently, several results have emerged independently in the literature with spectra having some type of quadratic form. In this note, we review those main results. Using a lower triangular matrix based on the Pascal’s triangle, we present an alternative unified approach to them. Ultimately, we provide a simple proof for Sylvester’s determinant claim. A comprehensive list of the major historical advances and generalizations, as well as the most recent contributions, is provided at the end.