Determinants and Inverses of Banded Toeplitz Matrices over \(\mathbb{F}\)\(_{{p}}\) Are Periodic

Abstract. Banded Toeplitz matrices over [Formula: see text], a well-known class of matrices, have been extensively studied in coding theory and automata theory. In this paper, we discover that determinants and inverses of banded Toeplitz matrices over [Formula: see text] exhibit periodicity. For matrices with bandwidth [Formula: see text], the period [Formula: see text] is determined by [Formula: see text] and the band coefficients. When [Formula: see text], the determinant becomes periodic as [Formula: see text] increases, while the inverse exhibits a block-periodic structure characterized by three prototype blocks that repeat on, above, and below the diagonal. Based on these results, we compute determinants in [Formula: see text] and inverses in [Formula: see text] and solve linear systems in [Formula: see text] over [Formula: see text]. Since [Formula: see text] and [Formula: see text] are typically small constants, our method is effectively [Formula: see text] with respect to the matrix order. To the best of our knowledge, this is the first time that the periodicity of determinants and inverses of general banded Toeplitz matrices over [Formula: see text] has been computed and proven.