A Note on the Square Roots of a Class of Circulant Matrices

Abstract

We prove that any $k$-circulant matrix and any even order skew $k$-circulant matrix are diagonalizable for any $k\in ℂ$. Then, we propose two algorithms for computing the square roots of the $k$-circulant matrix and the skew $k$-circulant matrix, respectively. In particular, we show that the square roots of the $k$-circulant matrix are still $k$-circulant matrices. Both the theoretical analysis and the numerical experiments show that our algorithms are faster than the standard Schur method.