Banded matrices and difference equations

Abstract

In this paper we consider discrete Sturm–Liouville eigenvalue problems of the form L(y) k:=∑ n μ=0(−Δ) μ r μ(k)Δ μy k+1−μ =λy k+1 for 0⩽k⩽N−n with y 1−n=⋯=y 0=y N+2−n=⋯=y N+1=0 , where N and n are integers with 1⩽n⩽N and under the assumption that r n(k)≠0 for all k. These problems correspond to eigenvalue problems for symmetric, banded matrices A∈ R (N+1−n)×(N+1−n) with bandwidth 2n+1. We present the following results: 1. an inversion formula, which shows that every symmetric, banded matrix corresponds uniquely to a Sturm–Liouville eigenvalue problem of the above form; 2. a formula for the characteristic polynomial of A , which yields a recursion for its calculation; 3. an oscillation theorem, which generalizes well-known results on tridiagonal matrices. These new results can be used to treat numerically the algebraic eigenvalue problem for symmetric, banded matrices without reduction to tridiagonal form.