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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

*Last edited by JaneFairfax (2009-03-28 00:31:32)*

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Quite fascinating Jane. That's the square root of the discriminant for a polynomial with roots a1, ..., an. Nice find.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Yes, its a nice find.

There are a couple of proofs of the result on the French Wikipedia website: http://fr.wikipedia.org/wiki/Matrice_de_Vandermonde. (Note that the website uses the transpose of the matrix I posted instead.)

The result is easily verified for small values of *n*. Suppose the result is true for some natural number *n* and consider the (*n*+1)×(*n*+1) matrix.

By multiplying the *i*th row by

By the inductive hypothesis, the determinant of the smaller matrix is

.That completes the proof by induction.

*Last edited by JaneFairfax (2009-03-28 02:33:49)*

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**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

The LHS is used by John E. Humphreys in *A Course in Group Theory* in the treatment of odd and even permutations.

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