Determinants and Inverses of banded Toeplitz Matrices over $\mathbb{F}_p$ Are Periodic
Chen Wang, Chao Wang
TL;DR
This work addresses computing determinants and inverses of banded Toeplitz matrices over finite fields $\mathbb{F}_p$ and reveals a structural periodicity tied to the band parameters via a linear feedback shift register framework. It derives a determinant formula and an efficient, $T$-driven inversion procedure, achieving $O(k^4)$ time for determinants and $O(k^5)+3kP(f)^2$ for inverses, with $P(f) \le p^{k-1}-1$ and the inverse expressible as three $P(f)\times P(f)$ blocks. The period of the determinant is $\mathrm{lcm}(p-1, P(f))$, and the inverse inherits a block-periodic structure with $T^{P(f)}=E$. These results enable scalable computations for high-order matrices with small bandwidths, offering practical impact for coding theory and automata applications that rely on banded Toeplitz structure over finite fields.
Abstract
Banded Toeplitz matrices over $\mathbb{F}_p$, as a well-known class of matrices, have been extensively studied in the fields of coding theory and automata theory. In this paper, we discover that both determinants and inverses of banded Toeplitz matrices over $\mathbb{F}_p$ exhibit periodicity. For a Toeplitz matrix with bandwidth $k$, The period $P(f)$ is related to the parameters on the band and is independent of the order, with an upper limit of $P(f) \le p^{k-1}-1$. We provide an algorithm which can compute the determinant of any order banded Toeplitz matrix within $O(k^4)$. And its inverse can be represented by three submatrices of size $P(f)*P(f)$ located respectively on the diagonal, above the diagonal, and below the diagonal. Thus, the computational cost for calculating the inverse is fixed, and our algorithm can solve it within $O(k^5)+3kP(f)^2$. This is the first time that the periodicity of determinants and inverses of general banded Toeplitz matrices over $\mathbb{F}_p$ has been computed and proven.