Explicit inverse of nonsingular Jacobi matrices
This work provides a theoretical foundation for inverting tridiagonal matrices, which is relevant for numerical linear algebra and differential equations, but the approach is incremental as it builds on known connections to difference equations.
The paper derives necessary and sufficient conditions for the invertibility of nonsingular Jacobi matrices and provides an explicit formula for their inverse using techniques from Sturm-Liouville boundary value problems and discrete Schrödinger operators.
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 Schrödinger 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.