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arXiv · 1810.09893

On nonsingularity of circulant matrices

Zhangchi Chen
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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 $2k+1$ is either a power of a prime, or a product of two distinct primes. For any other integer $2k+1$ we construct circulant matrices having determinant $0$. The smallest singular matrix appears when $2k+1=45$. The possibility for such matrices to be singular is rather low, smaller than $10^{-4}$ in this case.

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Zhangchi Chen. 2018-10-23. On nonsingularity of circulant matrices. https://doi.org/10.1016/j.laa.2020.12.010

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