## Circulant Determinant

Gradshteyn and Ryzhik (1970) define circulants by

 (1)

where is the th Root of Unity. The second-order circulant determinant is
 (2)

and the third order is

 (3)

where and are the Complex Cube Roots of Unity.

The Eigenvalues of the corresponding circulant matrix are

 (4)