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For a vector space or more generally a -module, then a quadratic form on is a function
which is homogeneous of degree 2 in that for all ,
and such that the polarization of
is a bilinear form.
Written entirely in terms of , the axioms for a quadratic form are:
(Besides the homogeneity, these come from two requirements of a bilinear form to preserve scalar multiplication and addition, respectively.) So we may alternatively define a quadratic form to be a map satisfying these three axioms.
A more general quadratic map (or homogeneous quadratic map to be specific) between vector spaces and is a map that satisfies the above three conditions. (Then an affine quadratic map is the sum of a homogeneous quadratic map, a linear map, and a constant, just as an affine linear map is the sum of a linear map and a constant.)
From the converse point of view, is a quadratic refinement of the bilinear form . (This always exists uniquely if is invertible, but in general the question involves the characteristic elements of . See there for more.)
Quadratic forms with values in the real numbers are called positive definite or negative definite if or , respectively, for all . See definiteness for more options.
The theory of quadratic forms emerged as a part of (elementary) number theory, dealing with quadratic diophantine equations, initially over the rational integers
The terminology “form” possibly originated with:
(which is cited as such in Gauss 1798, paragraph 151).
First classification results for forms over the integers were due to:
(which speaks of formas secundi gradus)
Hermann Minkowski, Grundlagen für eine Theorie der quadratischen Formen mit ganzzahligen Koeffizienten, Mémoires présentés par divers savants a l’Acad´emie des Sciences de l’institut national de France, Tome XXIX, No. 2. 1884.
Hermann Minkowski, Untersuchungen über quadratische Formen. Bestimmung der Anzahl verschiedener Formen, die ein gegebenes Genus enthält, Königsberg 1885; Acta Mathematica 7 (1885), 201–258
Discussion in the generality of noncommutative ground rings:
See also
Textbook accounts:
Richard Elman, Nikita Karpenko, Alexander Merkurjev, Algebraic and Geometric Theory of Quadratic Forms, Colloquium Publication 56, AMS (2008) [ams:coll-56, pdf]
Igor R. Shafarevich, Alexey O. Remizov: §6 in: Linear Algebra and Geometry (2012) [doi:10.1007/978-3-642-30994-6, MAA-review]
Course notes:
On the relation between quadratic and bilinear forms (pdf)
Bilinear and quadratic forms (pdf)
section 10 in Analytic theory of modular forms (pdf)
Quadratic refinements of intersection pairing in cohomology is a powerful tool in algebraic topology and differential topology. See
See also
Wikipedia, Quadratic form
Wikipedia, Definite quadratic form
Last revised on October 25, 2023 at 07:28:12. See the history of this page for a list of all contributions to it.