# Determinant of a general circulant matrix

I'm dealing with a problem that is comparable to "How do I calculate the circulant determinant $C(1, a, a^2, a^3,\dots , a^{n-1})$?", yet slightly more difficult: I was asked to determine the determinant of a general circulant matrix, i.e.

\begin{vmatrix} a_{1} & a_{2} & ... & a_{n-1} & a_{n} \\ a_{n} & a_{1} & ... & a_{n-2} & a_{n-1} \\ a_{n-1} & a_{n} & ... & a_{n-3} & a_{n-2} \\ \vdots & & & & \vdots \\ a_{2} & a_{3} & ... & a_{n} & a_{1} \end{vmatrix}

The answer is given, it contains the complex roots of 1, but I do not know how to attain these results.

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You have everything you need here. Eigenvalues with canonical eigenvectors. Just check that they do what they are supposed to do. Then take the product of the eigenvalues to get the determinant. – 1015 Apr 1 '13 at 13:34
I find it little satisfactory to insert the given expressions in the equations. May there be any way to construct these eigenvalues and -vectors instead? Thanks anyway – Francis Ryckaert Apr 1 '13 at 14:32