Determinants and Inverses of k-diagonal Toeplitz Matrices Are Periodic over F_p

This paper presents explicit formulas and algorithms for the determinants and inverses of k-diagonal Toeplitz matrices. We have achieved a computational complexity of O(k³ log n+k⁵)+kn² for their inverses. By considering the general case of k, we ensure versatility while remaining explicit and efficient compared to other algorithms. Furthermore, thanks to methodological innovation, we discovered that the determinants and inverses of k-diagonal Toeplitz matrices exhibit periodicity over finite fields. Over Fₚ, the inverses of nth-ordered k-diagonal Toeplitz matrices can be solved in O(k³ log c+k³ log p+k⁵)+3kc² where c is the period. This complexity is independent of n and therefore makes the computation of k-diagonal Toeplitz matrices over Fₚ effortless.