quadratic form (original) (raw)

Definition

A homogeneous polynomial of degree 2 in M is called a quadratic form (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 ℤ⁢[X,Y], X2-5⁢X⁢Y is a quadratic form, while Y3+2⁢X⁢Y and X2+Y2+1 are not.

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

where ai⁢j∈R.

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

For example, the quadratic form X2-5⁢X⁢Y can be rewritten as

Again, in the example of X2-5⁢X⁢Y, over ℚ it can be written as

However, it is not possible to represent X2-5⁢X⁢Y over ℤ 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:M→N be the (unique) evaluation homomorphism of the embedding, with ϕ⁢(Xi)=Xi for i<n and ϕ⁢(Xn)=0. Then for any quadratic form Q∈M, ϕ⁢(Q) is a quadratic form in N.

In particular, if we take N=R, and 𝐬=(s1,…,sn) with si∈R. Then the evaluation homomorphism ϕ at 𝐬 for any quadratic form Q∈M 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 Q∈M be a qudratic form. Take the direct sum of n copies of R and call this V. Define a map q:V→R 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:M→R, one can find a corresponding quadratic form Q∈M 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 𝐗T⁢𝐀𝐗, where 𝐀={ai⁢j} 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⁢(M⁢v), for all v∈Rn. The definition of equivalent quadratic forms is well-defined and it is not hard to see that this equivalence is an equivalence relation.

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

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

In the case where R=ℝ is the field of real numbers (or any formally real field), we say that a quadratic form is positive definite, negative definite, or positive semidefinite 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=𝐗T⁢𝐀𝐗 and Q2=𝐘T⁢𝐁𝐘, with 𝐗=(X1,…,Xm)T and 𝐘=(Y1,…,Yn)T. Then

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

Expressed in terms of Q1 and Q2, we write Q=Q1⊕Q2. For example, if Q1=5⁢X12+6⁢X22 and Q2=10⁢X1⁢X2, then

not 5⁢X12+6⁢X22+10⁢X1⁢X2(=Q1+Q2).

References

• 1 T. Y. Lam, Introduction to Quadratic Forms over Fields, American Mathematical Society (2004)