Talk:Gram–Schmidt process

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Does anyone know when the Gram Schimdt process was discovered? And who Mr. Gram and Mr. Schmidt are?

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Jørgen Pedersen Gram (1850-1916), Danish mathematician.

Sebastjan

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I was going to rewrite the process in terms of projections to clean it up a bit, but I'll probably stuff it up somewhere and be shouted at. So, if you feel up to doing so, I think it'd clear up the process greatly. Dysprosia 01:41, 26 Aug 2004 (UTC)

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I changed the calculation of u_2 etc. to use projection onto u_1 instead of onto e_1. This allows the process to work in fields where you cannot take square roots, such as the rationals. I have a program that produces a series of orthogonal vectors spanning the right space. norm@cap-lore.com .

Ewwwwww, I really don't like the notation proj_u v. If anything it should be written proj_v u, to emphasize that the projection is just a scaling of u.

Contents 1 Maybe we need an illustration 2 Basis 3 Notation 4 Factual accuracy on the method presented here

[edit] Maybe we need an illustration

Maybe we need an illustration of the projection just like most text books.

I just created this…

…would it fulfill its purpose? –Gustavb 07:16, 18 February 2006 (UTC)

Yes, I guess that is basically it. However, I think that u₂ is too long. We need to have

so the direction of u₂ is correct, but it should only go up to the dashed line. Does that make sense to you? If not, I'll change the picture myself, but I don't know the TeX package you are using (it looks quite easy to use though). -- Jitse Niesen (talk) 14:52, 18 February 2006 (UTC)

Yes, of course…guess I was tired when I did it. I have now created a new one but unfortunately I can't upload it due to a restriction in wikimedia commons which only allows users registered for more than four days to replace files (I registered yesterday.) –Gustavb 15:24, 18 February 2006 (UTC)

That seems rather silly to me; what's the purpose of prohibiting somebody from overwriting their own files? If you want, you can email it to me at my-first-name@ma.hw.ac.uk (making the obvious substitution) and I'll upload it on your behalf. You can also just wait a couple of days. -- Jitse Niesen (talk) 15:49, 18 February 2006 (UTC)

I don't think it checks who uploaded the original file – it just block all uploads for the purpose of reducing vandalism (I assume). So yes, it's a bad solution. Thanks for uploading the new version. –Gustavb 16:27, 18 February 2006 (UTC)

[edit] Basis

Does the set of vectors v₁, …, v_k not need to form a basis over the vector space? Tis is not mentioned anywhere on the page and is very important to the defintion I have learnt.Stb20 12:48, 22 January 2007 (UTC)

I'm rather sure that they don't need to form a basis. Do you have a text book which says that they do need to form a basis?

They do need to be linearly independent though. This is mentioned in the article. -- Jitse Niesen (talk) 11:35, 23 January 2007 (UTC)

My book does mention it yes, although it is refering to this process in forming an othogonal basis, rather than just an orthogonal set of vectors, so I assume this is the only time it applies. Is it still worth a mention? Stb20 11:47, 24 January 2007 (UTC)

[edit] Notation

for the projection, shouldn't it be proj(V) = [(V*U)/(U*U)] * U?? * is the dot product.

The author wrote it as (v,u)/(u, u) * u, which I don't understand what they mean.

<V,U> is a matrix or something else?

If it is a matrix then there shouldn't be division. If the expression is still correct, please explain what do the symbols mean.

Xfliang 05:40, 16 February 2007 (UTC)

<v,u> is another notation for the dot product of v and u. I added a bit to the article to explain this. -- Jitse Niesen (talk) 07:17, 16 February 2007 (UTC)

[edit] Factual accuracy on the method presented here

Maybe I'm reading the information on the page wrong, but as far as I can tell if I try and use the information here to create the 4th Legendre polynomial starting from 1,x,x^2,x^3 etc. I get an incorrect answer , however if I use the information on Mathworld I get the right answer. Eraserhead1 23:31, 18 February 2007 (UTC)

I'm sorry, but I can't see where the algorithm on this page is different from the algorithm in MathWorld. Could you perhaps explain where the algorithms differ? Is P₄ the first case where you get different results? -- Jitse Niesen (talk) 13:01, 10 March 2007 (UTC)