Talk:Vandermonde matrix

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I switched to the transpose, since the equation for polynomial interpolation was incorrect and the transposed matrix works better in companion matrix as well.

Something is wrong with the formula for the confluent Vandermonde matrix; the index surely cannot be 0? AxelBoldt 18:05, 11 Jul 2004 (UTC)

Now fixed. -- Jitse Niesen 13:36, 12 Jul 2004 (UTC)
Do we really want j<k vs. j≥k rather than j≤k vs. j>k in the case distinctions? Not knowing much of anything about these matrices, the latter seems more naturally to me, seeing that k=0 is the case of an ordinary Vandermonde matrix, and j=k gives a (-1)! in the denominator. AxelBoldt 16:17, 12 Jul 2004 (UTC)
You're absolutely right. Thanks for spotting this. -- Jitse Niesen 16:33, 12 Jul 2004 (UTC)

In the formula for the determinant, what is the set Sn? It is not defined or referenced. -- Grubber

This refers to the formula
\det(V) = \prod_{1\le i<j\le n} (\alpha_j-\alpha_i) =\sum_{\sigma \in S_n} \sgn (\sigma) \alpha_1^{\sigma(1)-1} ... \alpha_n^{\sigma(n)-1}
Sn denotes the set of permutations of {1, 2, ..., n}, and sgn(σ) denotes the signature of the permutation σ. However, I now removed the second expression for the determinant, since it is just a straightforward application of the Leibniz formula for the determinant, and it does not seem very useful. Anyway, thanks for bringing this to our attention. -- Jitse Niesen 03:55, 15 Jun 2005 (UTC)
I object to removing this. Why? To you, its a "straightforward application of the Leibniz formula". However, I know that AxelBoldt is very smart, and clearly, it wasn't straightforward to him. If it had been obvious, he would not have needed to ask "what is S_n?" I know AxelBoldt is smarter than 98% of all other wikipedia math readers, so I think its a bit of a mistake that average readers will have the Leibniz formula pop into their heads. Please, the goal of WP hould not be to be terse and concise; it should be healpful and useful. We should put the formula back in, and explain what S_n is, and explain that its obvious. (I mean, it is obvious, but sometimes one can be tired and bleary when reading WP, and when one is tired, its helpful to have things pop up). (Besides, the vandermonde determinant is all about the representation theory of the symmetric group, it makes sense to have the symmetric group appear here.). linas 05:06, 15 Jun 2005 (UTC)
If someone wouldn't mind keeping it and providing a reference and a definition, I'd appreciate it. I ran across Vandermonde matrices in a coding paper I'm reading, and all the 'obvious' tricks and formulae you all know are helpful! -- Grubber 09:35 15 Jun 2005 (UTC)
Fair enough, I put the formula back in. My primary reason for removing it was not that it's a "straightforward application of the Leibniz formula", but that I did not see that it is useful. However, I know little representation theory, so I believe you when you say it is useful in that context. -- Jitse Niesen 08:49, 15 Jun 2005 (UTC) (via edit conflict with Grubber)

[edit] Order of material

COMMENT BY ANOTHER PERSON: In its current incarnation, "the polynomial interpolation problem is ill-posed" comes before the mention of polynomial interpolation as an application. This needs to be switched around. --anon

I thought about it, and decided against it. The way things are now, the polynomial interpolation is used as just a motivation for introducting confluent Vandermonde matrices, so people can ignore that motivational sentence. But I did clarify that polymial interpolation is defined a few sections below. Oleg Alexandrov (talk) 01:09, 25 May 2006 (UTC)
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