Circulant matrix

A square matrix in which the rows are successive cyclic shifts of the first. The term circulant may denote such a matrix or the determinant of such a matrix.

Let $C$ denote the $n \times n$ circulant matrix with entries $C_{12} = C_{23} = \cdots = C_{n-1,n} = C_{n1} = 1$ and all other entries zero. If $\zeta$ is an $n$-th root of unity then the vector $v_\zeta = (1,\zeta,\ldots,\zeta^{n-1})^\top$ is an eigenvector of $C$ with eigenvalue $\zeta$. Further, a general circulant with first row $(a_0, a_1, \ldots, a_{n-1})$ is equal to the polynomial $a(C) = a_0 I + a_1 C + \cdots + a_{n-1} C^{n-1}$. Hence all circulant matrices commute, and have $v_\zeta$ as a common eigenvector with corresponding eigenvalue $a(\zeta)$.

References

• Marcus, Marvin, Minc, Henryk A survey of matrix theory and matrix inequalities Dover (1969)[1964] ISBN 0-486-67102-X Zbl 0126.02404
• Muir, Thomas A treatise on the theory of determinants. Dover Publications (1960) [1933] ISBN 0-486-60670-8