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Schur Matrix


The p×p square matrix formed by setting s_(ij)=xi^(ij), where xi is a pth root of unity. The Schur matrix has a particularly simple determinant given by

 detS=epsilon_pp^(p/2),
(1)

where p is an odd prime and

 epsilon_p={1   if p=1 (mod 4); i   if p=3 (mod 4).
(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 2p+1 are given by 1, 3, 5, 105, 81, 6765, ... (OEIS A003112, Vardi 1991).

Denote the Schur matrix S_p with the first row and first column omitted by S_p^'. Then

 permS_p=ppermS_p^',
(3)

where perm denoted the permanent (Vardi 1991).


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References

Graham, R. L. and Lehmer, D. H. "On the Permanent of Schur's Matrix." J. Austral. Math. Soc. 21, 487-497, 1976.Landau, E. Elementary Number Theory. New York: Chelsea, 1958.Sloane, N. J. A. Sequence A003112/M2509 in "The On-Line Encyclopedia of Integer Sequences."Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 119-122 and 124, 1991.

Referenced on Wolfram|Alpha

Schur Matrix

Cite this as:

Weisstein, Eric W. "Schur Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchurMatrix.html

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