Explicit inverse of nonsingular Jacobi matrices

TL;DR

This paper derives necessary and sufficient conditions for the invertibility of nonsingular Jacobi matrices and provides explicit formulas for their inverses using boundary value problem techniques related to difference equations.

Contribution

It introduces a novel method to explicitly compute the inverse of Jacobi matrices via boundary value problem solutions and conditions for invertibility.

Findings

Derived invertibility conditions for Jacobi matrices.

Explicit formulas for the inverse matrix entries.

Connected solutions to boundary value problems with matrix inversion.

Abstract

We present here the necessary and sufficient conditions for the invertibility of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute their inverse. The techniques we use are related with the solution of Sturm-Liouville boundary value problems associated to second order linear difference equations. These boundary value problems can be expressed throughout a discrete Schr\"odinger operator and their solutions can be computed using recent advances in the study of linear difference equations. The conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix, whereas the solutions of the boundary value problems provides the entries of the inverse matrix.