The determinant, spectral properties, and inverse of a tridiagonal $k$-Toeplitz matrix over a commutative ring

TL;DR

This paper derives explicit formulas for the determinant, characteristic polynomial, eigenvectors, and inverse entries of tridiagonal $k$-Toeplitz matrices over any commutative ring, enabling efficient computations and broad applicability.

Contribution

It provides universal formulas for key matrix properties of tridiagonal $k$-Toeplitz matrices over any commutative ring, using combinatorial and elementary linear algebra techniques.

Findings

Formulas for determinant, characteristic polynomial, and inverse entries in terms of elementary ring operations.

Algorithms based on these formulas are computationally efficient, with logarithmic complexity.

Results outperform existing literature in computational efficiency and generality.

Abstract

A square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find universal formulas for the determinant, the characteristic polynomial, some eigenvectors, and the entries of the inverse of any tridiagonal $k$-Toeplitz matrix (in particular, of any tridiagonal matrix) over any commutative unital ring, expressed in terms of the elementary operations of the ring. The results are proven using combinatorial identities and elementary linear algebra. We conduct a complexity analysis of algorithms based on our formulas, showing that they are efficient, and we compare our results favourably with those found in the literature. Concretely, the determinant, the characteristic polynomial, and any entry of the inverse of a tridiagonal $k$-Toeplitz matrix of size $n$ can each be found with $O(\displaystyle\log\frac nk+k)$ operations, while an eigenvector can be determined with $O(n+k)$ operations.