关于循环矩阵的非奇异性
On nonsingularity of circulant matrices
摘要：
在通信理论和编码中，期望在第一行中具有$k$ 1和$k+1$ 0的某些循环矩阵是非奇异的。证明了当$2k+1$是素数的幂或两个不同素数的乘积时，这类矩阵总是非奇异的.对于任何其他整数$2k+1$，我们构造具有行列式$0$的循环矩阵。当$2k+1=45$。这种矩阵奇异的可能性相当低，在这种情况下小于$10^{-4}$。
英文摘要：
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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