Circulant Matrix

An $n\times n$ Matrix ${\hbox{\sf C}}$ defined as follows,

$${\hbox{\sf C}} = \left[\begin{array}{ccccc} 1 & {n\choose 1} & {n\choose 2} & \cdo... ...ts\\ {n\choose 1} & {n\choose 2} & {n\choose 3} & \cdots & 1\end{array}\right]$$

$$\mathop{\rm det}\nolimits ({\hbox{\sf C}}) = \prod_{j=0}^{n-1} [(1+\omega_j)^n-1],$$

where $\omega_0\equiv 1$, $\omega_1$, ..., $\omega_{n-1}$ are the $n$th Roots of Unity. Circulant matrices are examples of Latin Squares.

See also Circulant Determinant

References

Davis, P. J. Circulant Matrices, 2nd ed. New York: Chelsea, 1994.

Stroeker, R. J. ``Brocard Points, Circulant Matrices, and Descartes' Folium.'' Math. Mag. 61, 172-187, 1988.

Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 114, 1991.