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When working with polynomials and an arbitrary weight function, there is a recursive Gram-Schmidt orthonomalization technique. More details are provided below:

http://mathworld.wolfram.com/Gram-SchmidtOrthonormalization.html 70.162.89.24 (talk) 05:51, 30 August 2013 (UTC)[reply]

Another example or two is needed, particularly something in a function (ie functional analysis) setting. For example, some simple polynomial examples with a suitable inner product. For example,

${\displaystyle \mathbf {u} _{1}=1}$, ${\displaystyle \mathbf {u} _{2}=t}$, ${\displaystyle \mathbf {u} _{2}=t^{2},\dots }$.

One covers examples like this in introductions to signal processing, so I imagine it's quite important in certain engineering fields. Also, this will produce orthogonal or orthonormal polynomials, making a nice connection to special functions. For example on the interval ${\displaystyle [-1,1]}$ with the inner product ${\displaystyle \int f(t)g(t)dt}$, one recovers the Legendre polynomials.Improbable keeler (talk) 09:29, 18 January 2018 (UTC)[reply]

I have two complaints about this section!

First of all, the section defines some vectors e_i and then never mentions them again. Frankly, I have no idea what's going on there.

Secondly, the section contains the text: "Note that the expression for uk is a "formal" determinant, i.e. the matrix contains both scalars and vectors; the meaning of this expression is defined to be the result of a cofactor expansion along the row of vectors."

I highly doubt that this explanation will be the least bit enlightening to anyone who didn't already know what was going on, and I also doubt that the link would be very enlightening with a bit of additional context. (For instance, the text should indicate which row the Laplace expansion is being taken over - the last one - and then indicate the formula as a linear combination of the v's with coefficients combing from the corresponding minor determinants.) 2602:30A:C04C:5F30:805:108B:A61A:2175 (talk) 23:37, 3 August 2014 (UTC)[reply]

Francesco Caravelli: I have been trying very hard to find the determinant formula in the literature. There is no reference in the wiki page. Very frustrating! Update: I have found a similar formula in Gantmacher: Theory of matrices (1959) Volume 1, Pages 256-258.

— Preceding unsigned comment added by 204.121.137.208 (talk) 17:28, 16 March 2017 (UTC)[reply] 

The Matlab implementation for the Gram-Schmidt process is for a specific norm and inner product definition (here being the Standard Euclidean Inner Product and by it's extension the 2-norm). Should be updated to reflect that. — Preceding unsigned comment added by BlackMetalStats (talk • contribs) 00:19, 3 April 2017 (UTC)[reply]

The article originally said that the method had appeared in the work by both Laplace and Cauchy, citing the Cheney and Kincaid book. ^[1] But the book only mentions that Laplace was "familiar" with the method. I couldn't see a reference to Cauchy on Google Books, but I don't have a copy of the book. Arguably a better historical reference is needed.Improbable keeler (talk) 06:57, 18 January 2018 (UTC)[reply]

If we consider ${\displaystyle proj_{n}(v):={\frac {<n,v>}{<n,n>}}n}$, I proof the set you get isn't orthogonal to ech other. Let's proof:

For the field we consider Complex Numbers. If ${\displaystyle proj_{n}(v):={\frac {<n,v>}{<n,n>}}n}$ then we begin by compute ${\displaystyle u_{i}}$.

$${\displaystyle u_{1}=v_{1}}$$ $${\displaystyle u_{2}=v_{2}-proj_{u_{1}}(v_{2})}$$

Now see what happen when ${\displaystyle <u_{2},u_{1}>}$:

$${\displaystyle <u_{2},u_{1}>=<v_{2}-proj_{u_{1}}(v_{2}),u_{1}>=<v_{2}-proj_{v_{1}}(v_{2}),v_{1}>=<v_{2},v_{1}>-<proj_{v_{1}}(v_{2}),v_{1}>=<v_{2},v_{1}>-<{\frac {<v_{1},v_{2}>}{<v_{1},v_{1}>}}v_{1},v_{1}>}$$

Now the ${\displaystyle {\frac {<n,v>}{<n,n>}}}$ is number in field so we can get it out of bracket as define ^[1].

${\displaystyle <v_{2},v_{1}>-{\frac {<v_{1},v_{2}>}{<v_{1},v_{1}>}}<v_{1},v_{1}>=<v_{2},v_{1}>-<v_{1},v_{2}>}$ and that ISN'T Zero in Complex Vector Space; But if you write the correct form which is ${\displaystyle proj_{n}(v):={\frac {<v,n>}{<n,n>}}n}$ every think make sense.

Firouzyan (talk) 21:35, 15 May 2019 (UTC)[reply]

I believe in the above there's a missing complex conjugation.

${\displaystyle <{\frac {<v_{1},v_{2}>}{<v_{1},v_{1}>}}v_{1},v_{1}>=\left({\frac {<v_{1},v_{2}>}{<v_{1},v_{1}>}}\right)^{*}<v_{1},v_{1}>}$ I've edited accordingly (changing it back to the pre-2019 state, I think).

Scott Lawrence (talk) 03:37, 5 September 2024 (UTC)[reply]

Apologies; I've just realized that I've run into this difference of convention between physics and the rest of the world. Changing the page back, but I'll add a short note somewhere to save other poor souls the effort.

Scott Lawrence (talk) 03:52, 5 September 2024 (UTC)[reply]