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#1 2009-03-28 09:41:49
- JaneFairfax
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Determinant of a Vandermonde matrix
Last edited by JaneFairfax (2009-03-28 23:31:32)
#2 2009-03-28 11:02:39
- Ricky
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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..."
#3 2009-03-29 00:55:32
- JaneFairfax
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Re: Determinant of a Vandermonde matrix
Yes, it’s 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 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.
Last edited by JaneFairfax (2009-03-29 01:33:49)
#4 2009-04-13 08:38:11
- JaneFairfax
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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.