Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

It is known that for the tridiagonal Toeplitz matrix, having the main diagonal with constant a0 = 2 and the two first off-diagonals with constants a1 = −1 (lower) and a−1 = −1 (upper), there exists closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. The latter matrix corresponds to the well known case of the 1D discrete Laplacian, but with a little care the formulas can be generalized to any triple (a0, a1, a−1) of complex values. 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 also can be identified in closed form. To achieve this, ad hoc sampling grids have to be considered, in connection with a new symbol associated with the standard Toeplitz generating function. In the second part, we restrict our attention to the symmetric real case (a0, aω = a−ω real values) and we analyze the relations with the standard generating function of the Toeplitz matrix. 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 that of the symmetrically sparse tridiagonal case with real a0, aω = a−ω, ω = 2, 3, . . ., and also quantitatively in terms of that having monotone symbols, as those related to classical Finite Difference discretization of the operators (−1) ∂ 2q ∂x2q , where the case of q = 1 coincides with a0 = 2, a1 = a−1 = −1.