Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

Abstract

Summary It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a0 and 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.

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

“…Unfortunately, as already shown in [2,15,16], the expansion (1.5) is not always satisfied even for s = 1. Below we give two conditions which ensure that the expansion holds.…”

“…is the remainder (the error), which satisfies the inequality |E (q) j,n,α | ≤ C α h α+1 for some constant C α depending only on α and f. We note that in the scalar-valued case s = 1, several theoretical and computational results are available in support of the above expansion [2,5,6,9,[14][15][16], including also extensions to preconditioned matrices and matrices arising in a differential context [1,13].…”

“…, s} such that j = k. This version of the global condition is of course much simpler to verify. Moreover, in the case s = 1 it reduces to the monotonicity condition already used in the literature; see [2,5,6,9,15,16] and references therein.…”

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

“…• the analysis of the non monotone case and its relations with the study in [12] for the special case where f (θ) = 2 − 2 cos(ωθ), ω ≥ 2 integer, and g(θ) = 1; • the extension of the results by [1] to the preconditioned Toeplitz case and the study of its connection with the general expansion in (1); • the extension of the numerical and theoretical study to a multidimensional, block setting, with special attention to the matrices coming from the approximation of elliptic differential operators. Proof For the sake of simplicity, we assume that r is nondecreasing (the other case has a similar proof).…”

“…However, even if the eigenvalues lying in the non monotone region give raise to an irregular error pattern, it seems that there exists a kind of 'deformed' periodicity in the error, like it is formally proven, without deformations, for the eigenvalues of T n (f ), f (θ) = 2 − 2 cos(ωθ), ω ≥ 2 integer, and g(θ) = 1 (see [12]). The latter observation indicates that a more complete study of this 'deformed' periodicity has to be considered in the future.…”