quadratic space

A quadratic space (over a field) is a vector spaceMathworldPlanetmath V equipped with a quadratic formMathworldPlanetmath Q on V. It is denoted by (V,Q). The dimensionPlanetmathPlanetmath of the quadratic space is the dimension of the underlying vector space. Any vector space admitting a bilinear formPlanetmathPlanetmath has an induced quadratic form and thus is a quadratic space.

Two quadratic spaces (V1,Q1) and (V2,Q2) are said to be isomorphicPlanetmathPlanetmathPlanetmath if there exists an isomorphic linear transformation T:V1V2 such that for any vV1, Q1(v)=Q2(Tv). Since T is easily seen to be an isometry between V1 and V2 (over the symmetric bilinear formsMathworldPlanetmath induced by Q1 and Q2 respectively), we also say that (V1,Q1) and (V2,Q2) are isometric.

A quadratic space equipped with a regular quadratic formPlanetmathPlanetmath is called a regular quadratic space.

Example of a Qudratic Space. The Generalized Quaternion Algebra.

Let F be a field and a,bF˙:=F-{0}. Let H be the algebra over F generated by i,j with the following defining relations:

  1. 1.


  2. 2.

    j2=b, and

  3. 3.


Then {1,i,j,k}, where k:=ij, forms a basis for the vector space H over F. For a direct proof, first note (ij)2=(ij)(ij)=i(ji)j=i(-ij)j=-ab0, so that kF˙. It’s also not hard to show that k anti-commutes with both i,j: ik=-ki and jk=-kj. Now, suppose 0=r+si+tj+uk. Multiplying both sides of the equation on the right by i gives 0=ri+sa+tji+uki. Multiplying both sides on the left by i gives 0=ri+sa+tij+uik. Adding the two results and reduce, we have 0=ri+sa. Multiplying this again by i gives us 0=ra+sai, or 0=r+si. Similarly, one shows that 0=r+tj, so that si=tj. This leads to two equations, sa=tij and sa=tji, if one multiplies it on the left and right by i. Adding the results then dividing by 2 gives sa=0. Since a0, s=0. Therefore, 0=r+si=r. Same argument shows that t=u=0 as well.

Next, for any element α=r+si+tj+ukH, define its conjugatePlanetmathPlanetmathPlanetmath α¯ by r-si-tj-uk. Note that α=α¯ iff αF. Also, it’s not hard to see that

  • α¯¯=α,

  • α+β¯=α¯+β¯,

  • αβ¯=β¯α¯,

We next define the norm N on H by N(α)=αα¯. Since N(α)¯=αα¯¯=α¯¯α¯=αα¯=N(α), N(α)F. It’s easy to see that N(rα)=r2N(α) for any rF.

Finally, if we define the trace T on H by T(α)=α+α¯, we have that N(α+β)-N(α)-N(β)=T(αβ¯) is bilinear (linear each in α and β).

Therefore, N defines a quadratic form on H (N is commonly called a norm form), and H is thus a quadratic space over F. H is denoted by


It can be shown that H is a central simple algebra over F. Since H is four dimensional over F, it is a quaternion algebraPlanetmathPlanetmath. It is a direct generalizationPlanetmathPlanetmath of the quaternions over the reals


In fact, every quaternion algebra (over a field F) is of the form (a,bF) for some a,bF.

Title quadratic space
Canonical name QuadraticSpace
Date of creation 2013-03-22 15:05:55
Last modified on 2013-03-22 15:05:55
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Definition
Classification msc 15A63
Classification msc 11E88
Synonym non-degenerate quadratic space
Related topic QuadraticForm
Related topic QuaternionAlgebra
Defines norm form
Defines isomorphic quadratic spaces
Defines isometric quadratic spaces
Defines generalized quaternion algebra
Defines regular quadratic space