Gradshteyn and Ryzhik (2000) define the circulant determinant by
![|x_1 x_2 x_3 ... x_n; x_n x_1 x_2 ... x_(n-1); x_(n-1) x_n x_1 ... x_(n-2); | | | ... |; x_2 x_3 x_4 ... x_1|=product_(j=1)(x_1+x_2omega_j+x_3omega_j^2+...+x_nomega_j^(n-1)),](/images/equations/CirculantDeterminant/NumberedEquation1.svg) |
(1)
|
where
is the
th
root of unity. The second-order circulant determinant
is
![|x_1 x_2; x_2 x_1|=(x_1+x_2)(x_1-x_2),](/images/equations/CirculantDeterminant/NumberedEquation2.svg) |
(2)
|
and the third order is
![|x_1 x_2 x_3; x_3 x_1 x_2; x_2 x_3 x_1|=(x_1+x_2+x_3)(x_1+omegax_2+omega^2x_3)(x_1+omega^2x_2+omegax_3),](/images/equations/CirculantDeterminant/NumberedEquation3.svg) |
(3)
|
where
and
are the complex cube roots
of unity.
The eigenvalues
of the corresponding
circulant matrix
are
![lambda_j=x_1+x_2omega_j+x_3omega_j^2+...+x_nomega_j^(n-1).](/images/equations/CirculantDeterminant/NumberedEquation4.svg) |
(4)
|
See also
Circulant Matrix
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References
Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press,
pp. 1111-1112, 2000.Vardi, I. Computational
Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 114, 1991.Referenced
on Wolfram|Alpha
Circulant Determinant
Cite this as:
Weisstein, Eric W. "Circulant Determinant."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CirculantDeterminant.html
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