Invertibility of circulant matrices of arbitrary size

TL;DR

This paper establishes general sufficient conditions based on linear combinations of entries that guarantee the invertibility of rational circulant matrices of any size, including special cases like matrices over finite fields and 0-1 matrices.

Contribution

It introduces broad sufficient conditions for invertibility of circulant matrices of arbitrary size, extending previous results to more general cases and specific matrix families.

Findings

Conditions guarantee invertibility for matrices over finite fields

Conditions apply to 0-1 circulant matrices

Results cover matrices generated by primitive elements

Abstract

In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer coefficients. Our result is general enough to show the invertibility of circulant matrices with any size and arrangement of entries. For example, using these conditions, we show the invertibility of the family of circulant matrices with particular forms of integers generated by a primitive element in $\mathbb{Z}_p$. Also, using a combinatorial structure of these sufficient conditions, we show invertibility for circulant $0, 1$-matrices.