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quadratic form in nLab

Contents

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

Definition

For VV a vector space or more generally a kk-module, then a quadratic form on VV is a function

q:V→k q\colon V \to k

which is homogeneous of degree 2 in that for all v∈Vv \in V, t∈kt \in k

q(tv)=t 2q(v) q(t v) = t^2 q(v)

and such that the polarization of qq

(v,w)↦q(v+w)−q(v)−q(w) (v,w) \mapsto q(v+w) - q(v) - q(w)

is a bilinear form.

Written entirely in terms of qq, the axioms for a quadratic form are:

q(tv)=t 2q(v)q(t v) = t^2 q(v),
q(tv+w)+tq(v)+tq(w)=tq(v+w)+t 2q(v)+q(w)q(t v + w) + t q(v) + t q(w) = t q(v + w) + t^2 q(v) + q(w),
q(u+v+w)+q(u)+q(v)+q(w)=q(u+v)+q(u+w)+q(v+w)q(u + v + w) + q(u) + q(v) + q(w) = q(u + v) + q(u + w) + q(v + w).

(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 q:V→kq\colon V \to k satisfying these three axioms.

A more general quadratic map (or homogeneous quadratic map to be specific) between vector spaces VV and WW is a map q:V→Wq\colon V \to W 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, qq is a quadratic refinement of the bilinear form (−,−)(-,-). (This always exists uniquely if 2∈k2 \in k is invertible, but in general the question involves the characteristic elements of (−u,−)(-u,-). See there for more.)

Quadratic forms with values in the real numbers k=ℝk = \mathbb{R} are called positive definite or negative definite if q(v)>0q(v) \gt 0 or q(v)<0q(v) \lt 0, respectively, for all v≠0v \neq 0. See definiteness for more options.

References

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:

Leonhard Euler, Novae demonstrations circa divisors numerorum formae xx+nyyx x + n y y…, Acad. Petrop. recitata, Nov 20, 1775, published poshumously

(which is cited as such in Gauss 1798, paragraph 151).

First classification results for forms over the integers were due to:

Carl Gauss, section V of Disquisitiones Arithmeticae, 1798

(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:

John Milnor, Dale Husemöller, Quadratic forms, Appendix 1 of: Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete 73, Springer (1973) [doi:10.1007/978-3-642-88330-9]