Efficient Computation for Invertibility Sequence of Banded Toeplitz Matrices

TL;DR

The paper addresses efficient invertibility testing for $n$-order banded Toeplitz matrices with bandwidth $2k+1$ by reducing the problem to the invertibility of a small $k\times k$ matrix. It proves a central theorem that $M_n$ is invertible if and only if the derived matrix $W_n$ is invertible, and introduces an algorithm that computes the invertibility sequence with time complexity $5k^2n/2+kn$ and space complexity $3k^2$, reusing computations across successive $W_i$ to avoid $O(k^3n)$ costs. This yields a practical preprocessing step for solving linear systems and computing inverses of banded Toeplitz matrices and generalizes beyond tridiagonal or pentadiagonal cases. The work also situates the approach relative to analogous invertibility determinations in cellular automata over finite fields, underscoring potential cross-domain applicability and efficiency gains in numerical PDE discretizations.

Abstract

When solving systems of banded Toeplitz equations or calculating their inverses, it is necessary to determine the invertibility of the matrices beforehand. In this paper, we equate the invertibility of an $n$-order banded Toeplitz matrix with bandwidth $2k+1$ to that of a small $k*k$ matrix. By utilizing a specially designed algorithm, we compute the invertibility sequence of a class of banded Toeplitz matrices with a time complexity of $5k^2n/2+kn$ and a space complexity of $3k^2$ where $n$ is the size of the largest matrix. This enables efficient preprocessing when solving equation systems and inverses of banded Toeplitz matrices.

Theorems & Definitions (9)

• Definition 1
• Theorem 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof
• Corollary 3
• proof