nLab quadratic form



Linear algebra

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Basic facts




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

q:Vk q\colon V \to k

which is homogeneous of degree 2 in that for all vVv \in V, tkt \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:Vkq\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:VWq\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 2k2 \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=k = \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 v0v \neq 0. 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:

  • 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:

(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

  • Rudolf Scharlau, Martin Kneser’s work on quadratic forms and algebraic groups, 2007 (pdf)

Textbook accounts:

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

Last revised on October 25, 2023 at 07:28:12. See the history of this page for a list of all contributions to it.