Banded matrices and difference equations

AbstractIn this paper we consider discrete Sturm–Liouville eigenvalue problems of the formL(y)k:=∑nμ=0(−Δ)μrμ(k)Δμyk+1−μ=λyk+1for 0⩽k⩽N−n with y1−n=⋯=y0=yN+2−n=⋯=yN+1=0, where N and n are integers with 1⩽n⩽N and under the assumption that rn(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