quadratic form

In this entry, unless otherwise specified, R is a commutative ring with multiplicative identityPlanetmathPlanetmath 1 and M=R[X1,,Xn] is a polynomial ringMathworldPlanetmath over R in n indeterminates.


A homogeneous polynomialMathworldPlanetmathPlanetmath of degree 2 in M is called a quadratic formMathworldPlanetmath (over R) in n indeterminates. In general, a quadratic form (without specifying n) over a ring R is a quadratic form in some polynomial ring over R.

For example, in [X,Y], X2-5XY is a quadratic form, while Y3+2XY and X2+Y2+1 are not.

In general, a quadratic form Q in n-indeterminates looks like


where aijR.

Letting 𝐗=(X1,,Xn)T, and 𝐀={aij} the n×n matrix, then we can rewrite Q as


For example, the quadratic form X2-5XY can be rewritten as


Now suppose the characteristicPlanetmathPlanetmath of R, char(R)2. In fact, suppose that 2 is invertiblePlanetmathPlanetmathPlanetmathPlanetmath in R (its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath denoted by 12). Since XiXj=XjXi, define bij=12(aij+aji). Then bii=aii and bij=bji. Furthermore, if 𝐁={bij}, then 𝐁 is a symmetric matrixMathworldPlanetmath and


Again, in the example of X2-5XY, over it can be written as


However, it is not possible to represent X2-5XY over by a symmetric matrix.

Evaluating a Quadratic Form

It is not hard to see that, given a quadratic form Q in n indeterminates, setting one of its indeterminates to 0 gives us another quadratic form, in (n-1) indeterminates. This is an informal way of saying the following:

embed R into N=R[X1,,Xn-1]. Let ϕ:MN be the (unique) evaluation homomorphism of the embeddingMathworldPlanetmathPlanetmathPlanetmath, with ϕ(Xi)=Xi for i<n and ϕ(Xn)=0. Then for any quadratic form QM, ϕ(Q) is a quadratic form in N.

In particular, if we take N=R, and 𝐬=(s1,,sn) with siR. Then the evaluation homomorphism ϕ at 𝐬 for any quadratic form QM is called the evaluation of Q at 𝐬, and we write it ϕ𝐬(Q), or simply Q(𝐬) (since ϕ is uniquely determined by 𝐬). In this way, a quadratic form Q can be realized as a quadratic map, as follows:

Let QM be a qudratic form. Take the direct sumMathworldPlanetmathPlanetmathPlanetmath of n copies of R and call this V. Define a map q:VR by q(v)=Q(v). Then q is a quadratic map.

Conversely, if 2 is invertible in R (so that char(R)2 is clear), then given a quadratic map q:MR, one can find a corresponding quadratic form QM such that q(v)=Q(v), by setting


where ei and ej are coordinate vectors whose coordinates are all 0 except at positions i and j respectively, where the coordinates are 1. Then Q defined by 𝐗T𝐀𝐗, where 𝐀={aij} is the desired quadratic form.

Equivalence of Quadratic Forms

From the above discussion, we shall identify a quadratic form as a quadratic map.

Two quadratic forms Q1 and Q2 are said to be if there is an invertible matrix M such that Q1(v)=Q2(Mv), for all vRn. The definition of equivalent quadratic forms is well-defined and it is not hard to see that this equivalence is an equivalence relationMathworldPlanetmath.

In fact, if 𝐀1 and 𝐀2 are matrices corresponding to (see the definition sectionPlanetmathPlanetmathPlanetmath) the two equivalent quadratic forms Q1 and Q2 above, then 𝐀1=MT𝐀2M.

For example, the quadratic form X2-Y2 is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to XY over any ring R where 2 is invertible, with M=(1-111).

In the case where R= is the field of real numbers (or any formally real field), we say that a quadratic form is positive definitePlanetmathPlanetmath, negative definite, or positive semidefinitePlanetmathPlanetmath according to whether its corresponding matrix is positive definite, negative definite, or positive semidefinite. The definiteness of a quadratic form is preserved under the equivalence relation on quadratic forms.

Sums of Quadratic Forms

If Q1 and Q2 are two quadratic forms in m and n indeterminates. We can define a quadratic form Q in m+n indeterminates in terms of Q1 and Q2, called the sum of Q1 and Q2, as follows:

write Q1=𝐗T𝐀𝐗 and Q2=𝐘T𝐁𝐘, with 𝐗=(X1,,Xm)T and 𝐘=(Y1,,Yn)T. Then


where 𝐙=(𝐗,𝐘)=(X1,,Xm,Y1,,Yn)T, and 𝐀𝐁 is the direct sum of matrices 𝐀 and 𝐁.

Expressed in terms of Q1 and Q2, we write Q=Q1Q2. For example, if Q1=5X12+6X22 and Q2=10X1X2, then


not 5X12+6X22+10X1X2(=Q1+Q2).


  • 1 T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society (2004)
Title quadratic form
Canonical name QuadraticForm
Date of creation 2013-03-22 12:19:22
Last modified on 2013-03-22 12:19:22
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 45
Author rm50 (10146)
Entry type Definition
Classification msc 11E08
Classification msc 11E04
Classification msc 15A63
Related topic PositiveDefinite
Related topic NegativeDefinite
Related topic SymmetricBilinearForm
Related topic QuadraticSpace
Related topic ProofOfGaussianMaximizesEntropyForGivenCovariance
Related topic IsotropicQuadraticSpace
Defines equivalent quadratic forms
Defines sum of quadratic forms
Defines evaluation of a quadratic form