A circulant determinant is the determinant of a circulant matrix. For a positive
integer 
, it has the factorization (Gradshteyn and Ryzhik 2000,
pp. 1065-1070)
 |
(1)
|
where 
for 
, 2, ..., 
enumerates the distinct 
th roots of unity. The second-order
circulant determinant is
 |
(2)
|
and the third-order circulant determinant is
 |
(3)
|
where 
and 
are the two nonreal cube roots of unity.
The eigenvalues 
of the corresponding 
circulant matrix
are
 |
(4)
|
