On nonsingularity of circulant matrices

Zhangchi Chen

doi:10.1016/j.laa.2020.12.010

paper

On nonsingularity of circulant matrices

arXiv:1810.09893 · doi:10.1016/j.laa.2020.12.010

Abstract

In Communication theory and Coding, it is expected that certain circulant matrices having $k$ ones and $k + 1$ zeros in the first row are nonsingular. We prove that such matrices are always nonsingular when $2 k + 1$ is either a power of a prime, or a product of two distinct primes. For any other integer $2 k + 1$ we construct circulant matrices having determinant $0$ . The smallest singular matrix appears when $2 k + 1 = 45$ . The possibility for such matrices to be singular is rather low, smaller than $1 0^{- 4}$ in this case.

12 pages. To be published in Linear Algebra and Its Applications