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.

翻译：在求解带状Toeplitz方程系统或其逆矩阵时，需预先判定矩阵的可逆性。本文将带宽为$2k+1$的$n$阶带状Toeplitz矩阵的可逆性等价地转化为$k \times k$小矩阵的可逆性。通过采用特定设计的算法，我们计算一类带状Toeplitz矩阵的可逆序列，其时间复杂度为$5k^2n/2+kn$，空间复杂度为$3k^2$，其中$n$为最大矩阵的规模。该成果可在求解带状Toeplitz方程系统和逆矩阵时实现高效预处理。