The 
square
matrix formed by setting 
, where 
is a 
th root of unity. The Schur
matrix has a particularly simple determinant given
by
 |
(1)
|
where 
is an odd prime and
 |
(2)
|
This determinant has been used to prove the quadratic reciprocity theorem (Landau 1958, Vardi 1991). The absolute
values of the permanents of the Schur matrix of
order 
are given by 1, 3, 5, 105, 81, 6765, ... (OEIS A003112,
Vardi 1991).
Denote the Schur matrix 
with the first row and first column omitted by 
. Then
 |
(3)
|
where perm denoted the permanent (Vardi 1991).
