Talk:Vandermonde matrix

message: Here is a proof, taken from physicsforums.com.  If this does not violate copyright, perhaps someone should clean it up and add it to the Wikipedia page. 
"To prove the formula given in the article, replace the ith row by
1 x x^2 ... x^(n-1)
Then, take the determinant. The determinant is a function of x. Lets call it
V(x).
It is a polynomial in x of degree (n-1). Hence, it has (n-1) roots. Furthermore, V(a_i) is the value of the vandermonde determinant, and you know that letting x=a_j for any j between 1 and n(excluding i) makes the determinant 0. Hence, a_i is a root of the polynomial V(x). Thus, (a_i - a_j) is a factor in the expansion of V(a_i). Repeating this with each of the i rows tells us that that
det(V)= C product (a_j - a_i), where the product is taken over 1<=i<j<=n. C is a constant. The fact that C=1 follows from induction. To show that C=1, just consider the cofactor expansion along the last column and examine the coefficient of the highest power of a_n. This is again a vandermonde determinant. Hence, C is the same constant as the smaller Vandermonde determinant. Of course, you need to check that in the case n=2 that C=1." 68.51.75.84 (talk) 03:13, 30 November 2009 (UTC)
thread_title: Add in proof of determinant formula

message: 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)
thread_title: Order of material

message: I came across some more properties. Do you think they're woth to be
included?
a) the eigenvalues a1,a2,...an of a vandermonde-matrix of dimension n have the following property:
   all eigenvalues are irrational (roots of order n)
   a1 +  a2 + a3 + ... + an = integer
   a1*a2 + a1*a3 + ... + a2*a3 + a2*a4 + ... +a(n-1) *an = integer
   a1*a2*a3 + ... + a(n-2)*a(n-1)*an = integer
   ...
   a1*a2*a3...*an = integer = 1!*2!*3!*...*(n-1)!  
   a1^2 + a2^2 + ... an^2 = integer
   a1^3 + a2^3 + ... an^3 = integer
   ...
   a1^n + a2^n + ... an^n = integer
b) let V be the vandermonde matrix, S1 the lower-triangular stirling-matrix kind 1,
    - which starts with
        1
       -1   1
        1 -3/2  1/2
       -1 11/6  -1   1/6
        ...
   (the rows are scaled by the reciprocal factorials)
   P the lower triangular binomial-matrix, S1~ the transpose od S1,
   then
     V*S1~ = P 
--Gotti 17:17, 8 January 2007 (UTC)
thread_title:  more properties 

message: Definitely needs a citation for this being "more commonly known": 4 hits on Google Books, one of which disagrees completely with this definition and one of which differs in a more subtle way!  I believe that the likely correct definition of Moore matrix, per a paper of Noam Elkies is that over GF(q) the matrix M has
M_{i,j} = \alpha_i^{q^{j-1}}.  Richard Pinch (talk) 18:29, 16 July 2008 (UTC)
thread_title:  Moore matrix 

message: This matrix is of fundamental importance in the quantum hall effect, and forms part of the ansatz used as the wavefunction for quasi-particles in 2D. Perhaps a mention?  —Preceding unsigned comment added by 137.195.250.2 (talk) 10:20, 19 July 2009 (UTC)
thread_title:  The Quantum Hall effect / Anyons 

message: Is there a way for this article to appear in the Wikipedia search if someone searches for "van der monde" instead of "vandermonde?"  I did a wikipedia search for "van der monde" (thinking this was how it was spelled) and got no hits in the search. Brian Maurizi 72.51.124.202 (talk) 20:10, 21 April 2010 (UTC)
replies: text: Vandermonde currently redirects to Alexandre-Théophile Vandermonde which mentions both Vandermonde matrices and Vandermonde's identity. One could create another redirect van der monde to Alexandre-Théophile Vandermonde. I think this would be not be inconsistent with the spirit of WP:REDIRECT#KEEP. I am going to create that redirect. — Tobias Bergemann (talk) 14:15, 22 April 2010 (UTC)
thread_title:  Searching For this Article 

message: The sentence
"Note that the Vandermonde determinant is alternating in the entries, meaning that permuting the αi by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the determinant."
is absolutely correct.  However, it is a little misleading, because in fact this property is true for the determinant in general and has nothing to do with the Vandermonde matrix in particular.  This should probably be mentioned.  Brian Maurizi 72.51.124.202 (talk) 20:13, 21 April 2010 (UTC)
replies: text: Well, this is true about permuting rows or columns of the determinant. If I understand this correctly, this sentence speaks about permuting variables when we view the as a polynomial. (In the other words: I believe that the intention was to explain that "Vandermonde polynomial is an alternating polynomial".) --Kompik (talk) 08:36, 22 April 2010 (UTC)
thread_title:  Comment on the Introduction 

message: The nice properties of the Vandermonde matrix only hold if the entries of the matrix commute. I think this should be explicitly mentioned.
Doetoe (talk) 15:44, 5 July 2011 (UTC)
thread_title:  Mention that it is assumed that the entries commute 

message: Although the article is careful to note the determinant formula and sequelae are valid only in the square case, essentially nothing is presented about the non-square case.  Just as the square case for distinct arguments gives us the matrix for polynomial interpolation, the general case gives the normal equations for least square fit of polynomials (when multiplied by the transpose).  This is an important application and gives an occasion to remark on rank and condition number, properties that are intrinsic to the non-square case. Hardmath (talk) 21:01, 2 August 2011 (UTC)
thread_title:  The non-square case 

message: I have to read through a whole page of dense mathematical definitions and anecdotes before I am told in the Vandermonde matrix#Applications what it's all about: evaluating a polynomial at multiple points.  — Preceding unsigned comment added by 2A01:E35:8B11:FA90:1E6F:65FF:FE3E:10A9 (talk) 23:42, 6 November 2017 (UTC)
replies: text: I tried to improve the structure of the article. There is still some work left to do.--2A01:E35:8B11:FA90:1E6F:65FF:FE3E:10A9 (talk) 01:16, 7 November 2017 (UTC)
thread_title:  Typical Wikipedia-style Math Article 

message: I have removed the new proof recently added by  for the following reasons.
* Conflict of interest (see WP:COI): The proof is based on an article written by the editor.
* Original research (see WP:OR): The proof appears only in a single WP:primary source.
* The proof has serious flaws:
**L is asserted to be nonsingular, which is wrong if the x_i are not  all distinct (easy to fix for a specialist, but confusing for a normal reader). 
**It is supposed without proof, and certainly wrong, that over every basis, the matrix of L is the evaluation matrix of the basis elements.
D.Lazard (talk) 09:59, 30 June 2020 (UTC)
replies: text: Although definitively wrong, 's proof suggested me a proof that is so simple that it deserves to be added to the article. Although I do not know any source for it, it is so and natural that it cannot be new, and exists certainly in the literature. Many thanks to everyone who will be able to find a source. D.Lazard (talk) 16:03, 30 June 2020 (UTC)
thread_title:  Recent addition removed 

message: The introductory section contains this sentence:
"The identical term Vandermonde matrix was used for the transpose of the above matrix by Macon and Spitzbart (1958)."
This information is far too obscure and trivial for it to belong in the introduction. It is not even clear that this information belongs in the article at all. 2601:200:C000:1A0:E0B5:CA14:5879:A6C9 (talk) 16:59, 18 June 2022 (UTC)
replies: text: I agree, and I have fixed the sentence. A have also removed the confusing sentence about Fourier transform. D.Lazard (talk) 17:34, 18 June 2022 (UTC)
thread_title:  Extremely obscure information does not belong in the introduction 

message: Who is the audience of this article?  It is a list of features with no apparent order that could only be understood by someone who already has expertise in this subject. The article seems to have no utility for the expert who would already know this information.  It also seems to have no utility for someone who wants to learn the subject, because the beginning has unnecessary details that could not be understood by someone who is not an expert. 
I have used the Vandermonde matrix many times in my work as an engineer doing spatial computation.  I never really needed to know the information the author thought belonged in the beginning of this article.
The article really needs a simple example somewhere near the first paragraph.  It would be great if the example was taken from history.  The example should also have a figure that really demonstrates the basic ideas.  The reader can hold this image in their mind as they try to understand the more advanced ideas of the subject. Openmir (talk) 17:23, 1 May 2023 (UTC)
thread_title:  No Introduction 

message: The 1st proof uses the factor theorem, which Wikipedia only states for univariate or bivariate polynomials. As stated, it does not apply to the kind of multivariate polynomial that the Vandermonde determinant is. I think you could manually imitate the proof of the Factor theorem by substitutions. Maybe this is really an application of Hilbert's Nullstellensatz. --Svennik (talk) 16:10, 15 October 2023 (UTC)
replies: text: This may be considered as an application of the Nullstellensatz, but the Nullstellensatz is a deep theorem with a difficult proof, which is much too strong here. I have slightly edited the proof for showing that the factor theorem applies, and I have improved the link for the unique factorization property of multivariate polynomials. D.Lazard (talk) 17:14, 15 October 2023 (UTC)
replies: text: I've made some changes to the factor theorem article. Now it states that the factor theorem is true even when the coefficients form only a commutative ring, and not a field. I've also provided three proofs. Observe that a polynomial ring R[X_1,X_2,\dotsc,X_n] is isomorphic to R[X_1,X_2,\dotsc,X_{n-1}] [X_n], giving us a way of using the factor theorem for multivariate polynomials like the Vandermonde determinant. --Svennik (talk) 17:29, 15 October 2023 (UTC)

text: The proof relying on polynomial properties is now elementary, as it involves only the factor theorem. As it is synthetic, while the two other proofs requires rather complicate matrix operations, I suggest to move this proof again as the first proof. D.Lazard (talk) 17:18, 18 October 2023 (UTC)
replies: text:  Done. --Svennik (talk) 21:38, 18 October 2023 (UTC)
thread_title:  I don't currently accept the 1st proof