Talk:Quadratic form

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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) = ax² + 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.

Indeed quadradic forms must be homogeneous of degree 2 i.e. the sum of the exponents in each term must be 2.

Contents 1 Quadratic forms in Statistics 2 error in isotropic defn? 3 Start out simple 4 Diagram 5 Error in definiteness definition

[edit] 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, ε is a random vector whose entries are independent with variance σ², and y = g + ε, then . I think there is enough to warrant a separate article on this; are there any objections? Btyner 18:41, 27 November 2005 (UTC)

[edit] 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.

[edit] 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.

[edit] Diagram

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

[edit] 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

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