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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)


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#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, its 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. http://i188.photobucket.com/albums/z198/fairjane/Woohoo.gif

Last edited by JaneFairfax (2009-03-29 01:33:49)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#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. smile


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

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