Banded matrices and difference equations

In this paper we consider discrete Sturm–Liouville eigenvalue problems of the form for 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 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 , 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.