Talk:Quadratic form/Archive 1

Archive 1

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Perhaps it would also be helpful to add the case of one variable? I say this because most of the general public will deal only with this case in high school and the first year of college calculus. Like so:)

$F(x)=ax^{2}+bx+c$

However I don't feel quite comfortable enough mathematically to actually modify the article, so if someone smart would like to implement this, I'd enjoy that :)

Goodralph 10:15, 3 Mar 2004 (UTC)

That's not a quadratic form, but a quadratic function q.v.

Charles Matthews 11:07, 3 Mar 2004 (UTC)

Added comment to that effect to article. - dcljr 06:51, 24 Feb 2005 (UTC)

Indeed quadradic forms must be homogeneous of degree 2 i.e. the sum of the exponents in each term must be 2. — Preceding unsigned comment added by 69.37.198.94 (talk) 21:48, 23 November 2005 (UTC)

Quadratic forms in Statistics

We need a either a section or an article on the properties of quadratic forms used in statistics. There are about a half dozen important theorems about these. For example, if $g$ is a vector of constants, $\epsilon$ is a random vector whose entries are independent with variance $\sigma ^{2}$, and $y=g+\epsilon$, then $\operatorname {E} \left[y'Ay\right]=\sigma ^{2}\operatorname {tr} \left[A\right]+g'Ag$. I think there is enough to warrant a separate article on this; are there any objections? Btyner 18:41, 27 November 2005 (UTC)

Start out simple

This article is a confusing read. It should start out presenting some simple results about symmetric matrices. 80% percent of the readers will be looking for these results, so they should be presented first.

Very few readers will be interested in topology and number theory. Therefore these sections should be moved to a section near the end or possibly transferred to a separate article. — Preceding unsigned comment added by 203.200.55.101 (talk) 13:39, 28 October 2006 (UTC)

Diagram

There should be a diagram of the quadratic form of a matrix. —Ben FrantzDale 22:30, 19 January 2007 (UTC)

Error in definiteness definition

To me there appears to be an error in the leading principal minor definition of positive and negative definiteness:

According to Robert A. Adams Calculus - A complete course 6th Edition, Section 10.6 Theorem 8 (p. 579): (Where A symmetric n×n matrix, D_i denotes the principal leading minor of size i×i)

a) If D_i>0 for 1≤i≤n, then A is positive definite
b) If D_i>0 for even numbers i in {1,2,…,n}, and D_i<0 for odd numbers i in {1,2,…,n}, then A is negative definite …

b) above seems to conflict with the first statement concerning principal leading minors in the article (which says that A is negative definite if D_i<0 for each i)

Hopefully someone who knows these things more clearly than me can edit the article (I was actually just looking for a proof of (the unproven) theorem 8 in Adams' book) Tinwelinto 20:30, 7 March 2007 (UTC)

You're right. I just deleted the bit about negative definite matrices. It is not so hard to derive, if necessary, and seems not that important here. I also removed the statement "the real symmetric matrix is positive semidefinite if and only if it has all non-negative leading principal minors" in view of the counterexample

${\begin{bmatrix}0&0\\0&-1\end{bmatrix}}.$

All two leading principal minors are zero, but the matrix is not positive semidefinite. -- Jitse Niesen (talk) 08:19, 12 March 2007 (UTC)

Why does the field that the vector space is defined over have to be real for positive definiteness to hold? Surely the quadratic form on C^n given by the n x n identity matrix is positive definite. —Preceding unsigned comment added by 82.14.70.59 (talk) 23:03, 12 January 2008 (UTC)

Only for the finite-dimensional case

"Bilinear forms are the full tensor product $V^{*}\otimes V^{*}$,..." well, this is only if V is finite-dimensional, right? Otherwise we have to write $(V\otimes V)^{*}$ instead, correct? Commentor (talk) 04:50, 3 March 2008 (UTC)

being more careful (or picky or pedantic) about the definition

I think this article is good---it certainly informed me---but i have a nit with it.

That is, i think it is confusing to write expressions like

$F(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz$

and also write

$B(u,v)=Q(u+v)-Q(u)-Q(v)$

with the idea that Q and F are the same sorts of things (quadratic forms) while B is something different (a bilinear form).

If it can be done in a non-clumsy way, i think it would be better to make it very clear just what we think a quadratic form is (e.g., a certain kind of function of one vector-valued variable, if we are thinking of these things as functions rather than merely formal polynomials), and use notation which suggests this.

My complaint sounds petty because after all if you spend some time thinking about it the meaning of the article is clear.

But the article is intended for a very broad audience and is not for publication in a journal. So i think it would be good to give up some of the minimality in exchange for a little more precision.

It is a good article, and my criticism, even if it is correct, is small. —Preceding unsigned comment added by 99.23.189.19 (talk) 06:38, 10 April 2009 (UTC)

Clean-up

Aaagh, this article is wild! I've spent some time rectifying obvious flukes, such as real quadratic forms appearing at the very end, even after integral forms, and the definition way towards the bottom. But ultimately, it is just too sprawling a topic to squeeze into a single article ("В одну повозку впрячь не можно коня и трепетную лань"). Note that EOM has a separate article on binary quadratic forms. If, indeed, the target audience includes people other than hard-core algebraists and an occasional algebraic topologist, then the article fails miserably in explaining the most fundamental case of quadratic forms in n variables over a field (no vector spaces or even modules over a ring where 2 is not invertible, please!). I tried to give some flavor of the theory in the real case, but it's still too sparse. It would stand to reason to treat binary quadratic forms, quadratic forms over a field (or even real/complex quadratic forms and separately rational quadratic forms) and integral quadratic forms in separate articles. This would also be consistent with historical development and the majority of literature. Also, less attention to definitions/conventions and more to the substance would help. Arcfrk (talk) 05:27, 13 April 2009 (UTC)

Identities

Moved over from the main article: section "Identities". I cannot immediately see if we have an article where this section will fit, but it certainly doesn't fit in here.

Identities

Starting with one solution {y₁, y₂, y₃} of the diagonal form,

$ay_{1}^{2}+by_{2}^{2}+cy_{3}^{2}=0$

an infinite more can be given using the identity,

$ax_{1}^{2}+bx_{2}^{2}+cx_{3}^{2}=(ay_{1}^{2}+by_{2}^{2}+cy_{3}^{2})(az_{1}^{2}+bz_{2}^{2}+cz_{3}^{2})^{2}$

where the x_i are,

$[x_{1},x_{2},x_{3}]=[uy_{1}-vz_{1},uy_{2}-vz_{2},uy_{3}-vz_{3}]\,$, and,

$[u,v]=[az_{1}^{2}+bz_{2}^{2}+cz_{3}^{2},2(ay_{1}z_{1}+by_{2}z_{2}+cy_{3}z_{3})]$

for arbitrary {z₁, z₂, z₃}. Similarly, for quaternary diagonal forms,

$ax_{1}^{2}+bx_{2}^{2}+cx_{3}^{2}+dx_{4}^{2}=(ay_{1}^{2}+by_{2}^{2}+cy_{3}^{2}+dy_{4}^{2})(az_{1}^{2}+bz_{2}^{2}+cz_{3}^{2}+dz_{4}^{2})^{2}$

where the x_i are,

$[x_{1},x_{2},x_{3},x_{4}]=[uy_{1}-vz_{1},uy_{2}-vz_{2},uy_{3}-vz_{3},uy_{4}-vz_{4}]\,$, and,

$[u,v]=[az_{1}^{2}+bz_{2}^{2}+cz_{3}^{2}+dz_{4}^{2},2(ay_{1}z_{1}+by_{2}z_{2}+cy_{3}z_{3}+dy_{4}z_{4})]$

for arbitrary {z₁, z₂, z₃, z₄}, and so on for any number of addends x_i.

Arcfrk (talk) 23:42, 22 September 2009 (UTC)

References

Alternative definition when 2 is not invertible

My revision http://en.wikipedia.org/w/index.php?title=Quadratic_form&oldid=354625703#Quadratic_forms was undid. Here's the definition in the case of a vector space over field F with reference. It seems to me that they should correspond to the definition in the module case, these things almost always do. If not please provide a source, I'd like to see it :)

Q:V -> F is a quadratic form if

• Q(av)=a^2Q(v)
• b(u,v)=Q(u+v)-Q(u)-Q(v) is bilinear

(Taylor 1992, page 54)

When F has characteristic not equal to 2 then Q(v)=b(v,v)/2 (Taylor 1992, page 55)

• Taylor, Donald E. (1992), The Geometry of the Classical Groups, vol. 9, Berlin: Heldermann Verlag, ISBN 3-88538-009-9, MR 1189139 {{citation}}: Text "Sigma Series in Pure Mathematics" ignored (help)

PS: Given enough time to visit the library I can provide further reference if you have trouble acquiring Donald Taylor's book. —Preceding unsigned comment added by 98.30.181.0 (talk • contribs)

There is nothing wrong with the definition itself. I've reverted it mainly because it leads to a different notion of the associated bilinear form, as compared with the next section, so it was just a quick consistency fix. I've now looked at a couple of standard sources for the algebraic theory of quadratic forms (Scharlau and Pfister), and they treat the case of characteristic ≠2 first, and characteristic 2 separately and somewhat later, and the associated bilinear form is defined differently! Earlier I commented that it was unwise to go for the ultimate generality and discuss quadratic forms over rings; but now I see that even over the fields, separate definitions are necessary for char 2.

Some time ago I started to streamline the presentation in this article and (re)wrote the first few sections, but ran out of steam and haven't tidied up the rest. The "definition" section and the next section need to be be merged and cleaned up. Right now, they are inaccurate, confusing, and redundant. For this reason, any changes made now will likely be transitory anyway. Arcfrk (talk) 06:41, 10 April 2010 (UTC)

Okay seems perfectly reasonable to me. I will certainly have some time to help with the streamlining / improving of the article in the next couple months. Admittedly my focus on quadratic forms has been mainly in the characteristic 2 case over fields. It is a little discouraging, but not surprising, that the definitions are somewhat inconsistent. I'm sure if we pool our efforts we can bring this article into better shape without too much trouble. I suppose it probably would be wise if I created a wikipedia account :) —Preceding unsigned comment added by 98.30.181.0 (talk) 16:06, 10 April 2010 (UTC)

error in isotropic defn?

The definition of isotropic and anisotropic in this article appears to be reversed, at least to me. It defines an "isotropic space" as one whose form has a non-trivial kernel. Surely such a space should be anisotropic, instead? That defn has been there a longgg time. linas 13:11, 19 July 2006 (UTC)

The definition in the article seems to agree with J. P. Serre in "A Course in Arithmetic" (interesting title!), so Isuspect it might be right. Madmath789 14:57, 19 July 2006 (UTC)

The definition is correct. Isotropic is the standard name for spaces with non zero elements $v$ such that $Q(v)=0$. This is according to the refrence listed on the bottom of the article.

Perhaps this is a development more recent the Serre's book, but the definition I've seen for isotropic has to do with the the associated bilinear form:

• v is isotropic if b(v,v)=0

and the other definition is

• v is singular if Q(v)=0

Note in odd characteristic these definitions are the same because the quadratic form is completely determined by its' bilinear form. However, in even characteristic there are isotropic vectors that are not singular. These references may also handle the

char != 2 case first and then revamp it later. IMHO there is something very important missing here. —Preceding unsigned comment added by 98.30.181.0 (talk) 15:49, 16 April 2010 (UTC)