Circulant Determinant

Gradshteyn and Ryzhik (1970) define circulants by

$\begin{displaymath} \left\vert\matrix{x_1 & x_2 & x_3 & \cdots & x_n\cr x_n & x_... ... (x_1+x_2\omega_j+x_3{\omega_j}^2+\ldots+x_n{\omega_j}^{n-1}), \end{displaymath}$

(1)

where $\omega_j$ is the

th Root of Unity. The second-order circulant determinant is

$\begin{displaymath} \left\vert\matrix{x_1 & x_2\cr x_2 & x_1\cr}\right\vert=(x_1+x_2)(x_1-x_2), \end{displaymath}$

(2)

and the third order is

$\begin{displaymath} \left\vert\matrix{x_1 & x_2 & x_3\cr x_3 & x_1 & x_2\cr x_2 ... ..._3)(x_1+\omega x_2+\omega^2 x_3)(x_1+\omega^2 x_2+\omega x_3), \end{displaymath}$

(3)

where $\omega$ and $\omega^2$ are the Complex Cube Roots of Unity.

The Eigenvalues $\lambda$ of the corresponding $n\times n$ circulant matrix are

$\begin{displaymath} \lambda_j=x_1+x_2\omega_j+x_3{\omega_j}^2+\ldots+x_n{\omega_j}^{n-1}. \end{displaymath}$

(4)

See also Circulant Matrix

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1111-1112, 1979.

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