Banded Matrices and Difference Equations

In this paper we consider Sturm-Liouville eigenvalue problems of the formfor 0 ≤ ≤ − with = ⋯ = = = ⋯ = = 0, where and are integers with 1 ≤ ≤ and under the assumption that () ≠ 0 for all . These problems correspond to eigenvalue problems for symmetric, banded matrices with bandwidth 2 + 1. We present the following results: - an inversion formula, which shows that symmetric, banded matrix corresponds uniquely to a Sturm-Liouville eigenvalue problem of the above form - a formula for the characteristic polynomial of , which yields a for its calculation, and - an , 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