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Circulant Determinant

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)),
(1)

where omega_j is the nth 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),
(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),
(3)

where omega and omega^2 are the complex cube roots of unity.

The eigenvalues lambda of the corresponding n×n circulant matrix are

lambda_j=x_1+x_2omega_j+x_3omega_j^2+...+x_nomega_j^(n-1).
(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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