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

Russell Jay Hendel

TL;DR

This work resolves the question of when a Laplace-expansion procedure converges by proving convergence for any family of determinants derived from banded, square, Toeplitz matrices, with termination after $R$ expansions where $R$ is the Toeplitz order. It introduces a formal framework using operator notations, a Reduction Lemma, and a constructive Expand algorithm, and provides a complete inductive proof of convergence, illustrated by a concrete $R{=}3$ example that yields a linear recurrence of order $6$. The results generalize the Evans–Hendel approach to a broad class of determinant families and offer a systematic method to derive recurrences for determinants tied to graph- or matrix-pattern families, potentially impacting related spectral and combinatorial analyses.

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.

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

TL;DR

This work resolves the question of when a Laplace-expansion procedure converges by proving convergence for any family of determinants derived from banded, square, Toeplitz matrices, with termination after expansions where is the Toeplitz order. It introduces a formal framework using operator notations, a Reduction Lemma, and a constructive Expand algorithm, and provides a complete inductive proof of convergence, illustrated by a concrete example that yields a linear recurrence of order . The results generalize the Evans–Hendel approach to a broad class of determinant families and offer a systematic method to derive recurrences for determinants tied to graph- or matrix-pattern families, potentially impacting related spectral and combinatorial analyses.

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.
Paper Structure (11 sections, 2 theorems, 21 equations)

This paper contains 11 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Matrix Families
  4. The Reduction Lemma
  5. Illustrative Example
  6. Review of Solving Simultaneous Equations of Determinant Families
  7. Pseudocode and the Main Theorem
  8. Base Step and Induction Assumption
  9. Induction Step
  10. Completion of the Proof
  11. Comparison of Approaches

Key Result

Lemma 1

For a banded, square, Toeplitz, matrix family of Toeplitz order $R:$ (a) If for some $s, 1 \le s \le R-1,$$A_i=i=B_i, 1 \le i \le s$ then $(A;B) = (A',B'),$ with $C'_{i-s}=C_i - s, \text{ for } C \in \{A,B\} \text{ and } s+1 \le i \le R.$ If $A_i=i=B_i, 1 \le i \le R$ then $(A;B)=(0;0).$ We will the

Theorems & Definitions (4)

  • Lemma 1: Reduction
  • proof
  • Theorem 2: Main
  • Remark 3