Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a0and on the two first off-diagonals the constants a1(lower) and a-1(upper), which are all complex values, there exist closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. For example, for the 1D discrete Laplacian, this triple is (a0,a1,a-1)=(2,-1,-1). In the first part of this article, we consider a tridiagonal Toeplitz matrix of the same form (a0,aω,a-ω), but where the two off-diagonals are positioned ω steps from the main diagonal instead of only one. We show that its eigenvalues and eigenvectors can also be identified in closed form and that interesting connections with the standard Toeplitz symbol are identified. Furthermore, as numerical evidences clearly suggest, it turns out that the eigenvalue behavior of a general banded symmetric Toeplitz matrix with real entries can be described qualitatively in terms of the symmetrically sparse tridiagonal case with real a0, aω=a-ω, ω=2,3,..., and also quantitatively in terms of those having monotone symbols. A discussion on the use of such results and on possible extensions complements the paper.