# Singularity of Sparse Circulant Matrices is NP-complete - Computer Science > Computational Complexity

Abstract: It is shown by Karp reduction that deciding the singularity of $2^n - 1\times 2^n - 1$ sparse circulant matrices SC problem is NP-complete. We canwrite them only implicitly, by indicating values of the $2 + nn + 1-2$eventually nonzero entries of the first row and can make all matrix operationswith them. The positions are $0, 1, 2^{i} + 2^{j}$. The complexity parameter is$n$. Mulmuley-s work on the rank of matrices \cite{Mulmuley87} makes SC standalone in a list of 3,000 and growing NP-complete problems.

Author: Ilia Toli

Source: https://arxiv.org/