Proof of Convergence of a Laplace Expansion Algorithm For Calculating Recursions Satisfied by a Family of Determinants

TL;DR

This paper proves the convergence of a Laplace expansion algorithm for a broad class of determinants, extending previous work and improving the algorithm's efficiency in calculating recursions for Toeplitz matrices.

Contribution

It establishes the convergence of the algorithm for banded Toeplitz matrices and enhances the original method used by Evans and Hendel.

Findings

Proves convergence for arbitrary banded Toeplitz determinants

Improves the efficiency of the original Laplace expansion algorithm

Extends applicability to a wider class of matrices

Abstract

In Evan and Hendel's recent proof of an outstanding conjecture on the resistance distances of a family of linear 3-trees, a key technique in the proof was calculating the recursion satisfied by a family of determinants. The underlying algorithm employed to prove the conjecture converged (i.e. terminated) in the particular case studied, and the paper presented an open question on when such a procedure converges in general. This paper proves convergence of the procedure for an arbitrary family of determinants of banded, square, Toeplitz matrices. Moreover, the algorithm in this paper improves several aspects of the algorithm of Evans and Hendel.