World Library  
Flag as Inappropriate
Email this Article

Definite bilinear form

Article Id: WHEBN0000372624
Reproduction Date:

Title: Definite bilinear form  
Author: World Heritage Encyclopedia
Language: English
Subject: Differential geometry, Inner product space, Linear algebra, Parabola, Minkowski space, Pseudo-Riemannian manifold, Mercer's theorem, Split-complex number, Niemeier lattice, Mercer's condition
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Definite bilinear form

In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every nonzero vector of V. According to that sign, the quadratic form is called positive definite or negative definite.

A semidefinite (or semi-definite) quadratic form is defined in the same way, except that "positive" and "negative" are replaced by "not negative" and "not positive", respectively. An indefinite quadratic form is one that takes on both positive and negative values.

More generally, the definition applies to a vector space over an ordered field.[1]

Associated symmetric bilinear form

Quadratic forms correspond in one-to-one way to symmetric bilinear forms over the same space. A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations:

\, Q(x) = B(x,x)
\, B(x,y) = B(y,x) = \tfrac{1}{2} (Q(x + y) - Q(x) - Q(y))

Example

As an example, let V = ℝ2, and consider the quadratic form

Q(x)=c_1{x_1}^2+c_2{x_2}^2 \,

where x = (x1, x2) and c1 and c2 are constants. If c1 > 0 and c2 > 0, the quadratic form Q is positive definite. If one of the constants is positive and the other is zero, then Q is positive semidefinite. If c1 > 0 and c2 < 0, then Q is indefinite.

See also

References

  • Nathanael Leedom Ackerman (2006) University of California, Berkeley.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.