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#1 2009-03-27 10:41:49

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

Determinant of a Vandermonde matrix

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


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#2 2009-03-27 12:02:39

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

Re: Determinant of a Vandermonde matrix

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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#3 2009-03-28 01:55:32

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

Re: Determinant of a Vandermonde matrix

Yes, it’s a nice find. smile

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 ith row by

and subtracting from the (i+1)th row
  we have

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

.

That completes the proof by induction. Woohoo.gif

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


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#4 2009-04-12 10:38:11

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

Re: Determinant of a Vandermonde matrix

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


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