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Circulant Determinant

Gradshteyn and Ryzhik (1970) define circulants by

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

\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

\end{displaymath} (4)

See also Circulant Matrix


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.

© 1996-9 Eric W. Weisstein