Talk:Quadratic form

Quadratic forms defined by a non-symmetric matrix

Why my contribution was deleted without any attempt of consensus?

You are not encouraging new editors.

DIFF: http://en.wikipedia.org/w/index.php?title=Quadratic_form&action=historysubmit&diff=328339698&oldid=327825192

Arcfrk: if you find it "unhelpful/confusing", why didn't you ask first for a better solution. We can clarify or expand the remark. It is a sufficiently relevant fact.

What if a revert your change arbitrarily as you did? Wikipedia can't work that way.

Sorry for my bad english.

Francisco Albani (talk) 01:32, 29 November 2009 (UTC)

I would certainly prefer to see this discussed at symmetric bilinear form. It is clearer to understand the relationship between general and symmetric bilinear forms as one step, with the remark that the semi-sum symmetrization depends on being able to divide by 2 in the field; and then the relationship between symmetric bilinear forms and quadratic forms as another step. If you can divide by 2 freely there is no problem; but that is not always the case in this subject. Charles Matthews (talk) 09:25, 29 November 2009 (UTC)

This discussion clears up my confusion about the earlier statement, "there is a one to one correspondence between quadratic forms and symmetric matrices that determine them."

The article assumes the matrices are symmetric, without offering a reason. This should be fixed. Comfr (talk) 18:14, 29 June 2022 (UTC)

I have added some clarification. If some other assertion is confusing, you have to be more precise in the localization of the assertion in the article. D.Lazard (talk) 18:26, 29 June 2022 (UTC)

non-singular or non-degenerate quadratic form

I don't think non-singular is defined correctly for the quadratic form. Orthogonal_group defines the orthogonal group to be linear transformations that preserve some non-singular quadratic form. So I assume this is what's called a non-degenerate quadratic form in the literature I've read. Here's how I've seen it before:

• A quadratic form is non-singular (non-degenerate) if the kernel of the associated bilinear form is anisotropic. —Preceding unsigned comment added by Somethingcompletelydifferent (talk • contribs) 15:20, 11 April 2010 (UTC)

Well, since q(x) = b(x,x), the restriction of the quadratic form q to the kernel of b is identically zero. So the only way for ker b to be anisotropic is if it only contains zero vector, i.e. ker b = 0. Do you have a reference? Does it have anything to do with characteristic 2 or modules over a ring? In both cases, the definitions of the associate bilinear form and of "non-degenerate" need to be modified. Arcfrk (talk) 03:51, 13 April 2010 (UTC)

Unfortunately q(x)=b(x,x) only when char !=2. In characteristic 2 there is a difference. In fact there are bilinear forms in characteristic 2 that are not the associated bilinear form to any quadratic form :) —Preceding unsigned comment added by 98.30.181.0 (talk) 15:51, 16 April 2010 (UTC)

Sorry thought I was signed in above. Is my comment —Preceding unsigned comment added by Somethingcompletelydifferent (talk • contribs) 15:55, 16 April 2010 (UTC)

Well, since you agree that it's a characteristic 2 issue, why don't you write a section dedicated to char 2 case, as you previously indicated you wanted to? It is certainly needed for the completeness of coverage! Think it through and give all the necessary definitions and theorems. In the meantime, please, do not insert definitions/statements inconsistent with the rest of the text. All of the present section is dealing with characteristic NOT 2, as prominently displayed just a short while before. In particular, the term "associate bilinear form" has a precise meaning, q(x)=b(x,x), and your way of phrasing the definition of nonsingular simply does not work in this context. Arcfrk (talk) 06:02, 24 April 2010 (UTC)

"intentional" space

Arcrfk, you insist on keeping a space at the beginning of a line to set the following in typewriter text: "Let us assume that the characteristic of K is different from 2." If you feel strongly that it should be set in its own text box, fine, but there is no reason to set it in typewriter text. It is completely nonstandard within Wikipedia. I don't see why it can't just be its own line without anything special to set it off, but setting it typewriter text is just wrong. —Ben FrantzDale (talk) 14:15, 22 October 2010 (UTC)

geometric meaning

Can someone related the quadratic form to geometric meaning. when it is positive definite, it is ellipsoid? what condition corresponds to hyperboloid (elliptic or parabolic) or paraboloid???

Suppose the equation is $x^{T}Ax=1$, then it seems that

if the matrix can be turned into a diagonal matrix, then it is an ellipsoid or a hyperboloid. If all the eigen values are non negative, then it is an ellipsoid, if some eigen values are negative, then it is a hyperboloid. If the matrix can not be turned into a diagonal one, then it is a paraboloid (either elliptic or hyperbolic). If all the eigen values are non negative, then it is elliptic, if some eigen values are negative, then it is hyperbolic).

Jackzhp (talk) 14:55, 13 February 2011 (UTC)

"Jacobi's Theorem

This article mentions "Jacobi's Theorem" under "Real Quadratic Forms" but doesn't link that to anywhere and when I search for jacobi's theorem in wikipedia, I don't see a disambiguation link for this use of the term.

Bill Smith (talk) 19:17, 20 January 2013 (UTC)

It refers to this sentence a few lines earlier:

A fundamental theorem due to Jacobi asserts that q can be brought to a diagonal form

Deltahedron (talk) 19:49, 20 January 2013 (UTC)

Citations

I was not sure this article really must have additional citations. I think it was basic stuff from the authors point of view and not especially dubious at any points. But I admit I am new to this topic so that can be factored into my opinion. Further I would like to mention that over dependence on citations can be meaningless. For example, you can find a million citations that support significant man caused global warming but that may actually be negative evidence. Pardon the political reference and substitute any example where citations may not be accurate. However if there is something truly dubious here let's get to that issue. § — Preceding unsigned comment added by Berrtus (talk • contribs) 08:23, 15 April 2013 (UTC)

While lending credibility to the correctness of information is an important function of citations, that's actually secondary to the more fundamental use: showing the reader where the information appears. This is just an encyclopedia, so its role is to act as a reference to reputable sources (sometimes despite the sources' correctness or incorrectness.) In any case, having a citation to a good reference is invaluable to a reader who wants to learn more. Having the "needs more citations" flag doesn't always have to be interpreted as a black eye! I think in this case, the contents of the article is OK: however, it looks like someone wished they had some more detailed pointers to where they could continue reading :) Rschwieb (talk) 13:47, 15 April 2013 (UTC)

I have added various items as a Further Reading section. Deltahedron (talk) 16:51, 15 April 2013 (UTC)

What is a "form"?

Is form being used here to refer to a particular arrangement of something else, as in an ice cube is water in a frozen and cubic form? Or is form a thing in its own right, as in a form for shaping concrete? Or??? Gwideman (talk) 09:15, 22 February 2021 (UTC)

@Gwideman: A form in linear algebra refers to a scalar-valued function of vectors, usually linear or quadratic, and always a homogeneous polynomial. Further questions should be asked at WP:RDMA.--Jasper Deng (talk) 09:32, 22 February 2021 (UTC)

Why should questions about the wording of this page be directed to WP:RDMA? Gwideman (talk) 14:18, 22 February 2021 (UTC)

However, the previous first sentence may be confusing for some readers, and I have edited it per WP:LEAST for linking to the general definition (form (mathematics)). D.Lazard (talk) 09:46, 22 February 2021 (UTC)

That's a helpful improvement. So here form is unrelated to the form in sentences like "a polynomial written in factored form". Gwideman (talk) 14:16, 22 February 2021 (UTC)

Yes, in this sentence, "form" is a term of mathematical jargon that must be viewed as a synonym of "shape". D.Lazard (talk) 16:34, 22 February 2021 (UTC)

Generalization?

In the generalization section, it asserts that for any quadratic form, there exists an R-bilinear form b : M × M → R such that q(v) is the associated quadratic form.

Is this correct, for an arbitrary module M? I can only see how to do it when M is projective. It would be nice to have a citation.

Hasire (talk) 23:29, 22 July 2021 (UTC)

Quadratic spaces

Why introducing the map Q? What does it tell more than q does? Apparently Q=q. Madyno (talk) 10:40, 20 November 2022 (UTC)

I have rewritten the beginning of the section for clarifying that. Also, the section is called now § Quadratic space. D.Lazard (talk) 15:06, 20 November 2022 (UTC)

I don't understand. The quadratic form $q:V\to K$ is the same type of function as $Q:V\to K$; both accept as their arguments an element of $V$. If $V=K^{n}$ the elements of $V$ are the n-tuples $(v_{1},\ldots ,v_{n})\in K^{n}$, which has nothing to do with the standard basis. Then a value of $q$ is $q((v_{1},\ldots ,v_{n}))$. Madyno (talk) 17:21, 28 November 2022 (UTC)

Subsequent identical terms

@Anita5192 Hi, in the sentence:

The study of particular quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers

There exists two subsequent "particular" terms. This is not grammatically wrong, but it is better that one of them become "specially". Thanks, Hooman Mallahzadeh (talk) 14:42, 22 March 2023 (UTC)

 Fixed—Anita5192 (talk) 14:52, 22 March 2023 (UTC)