quadratic form

In this entry, unless otherwise specified, $R$ is a commutative ring with multiplicative identity $1$ and $M=R[X_1,\mathrm{\dots },X_n]$ is a polynomial ring over $R$ in $n$ indeterminates.

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 $\mathbb{Z}[X,Y]$, $X^2-5XY$ is a quadratic form, while $Y^3+2XY$ and $X^2+Y^2+1$ are not.

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

$$Q=a_{11}{X}_1^2+a_{12}{X}_1{X}_2+\mathrm{\cdots }+a_{n,n-1}{X}_n{X}_{n-1}+a_{nn}{X}_n^2=\sum _{1\le i,j\le n}{a}_{ij}{X}_i{X}_j$$

where $a_{ij}\in R$.

Letting $𝐗={(X_1,\mathrm{\dots },X_n)}^{\mathrm{T}}$, and $𝐀=\{a_{ij}\}$ the $n\times n$ matrix, then we can rewrite $Q$ as

$$Q={𝐗}^{\mathrm{T}}\mathrm{𝐀𝐗}.$$

For example, the quadratic form $X^2-5XY$ can be rewritten as

Now suppose the characteristic of $R$, $\mathrm{char}(R)\ne 2$. In fact, suppose that $2$ is invertible in $R$ (its inverse denoted by $\frac{1}{2}$). Since $X_i{X}_j=X_j{X}_i$, define $b_{ij}=\frac{1}{2}(a_{ij}+a_{ji})$. Then $b_{ii}=a_{ii}$ and $b_{ij}=b_{ji}$. Furthermore, if $𝐁=\{b_{ij}\}$, then $𝐁$ is a symmetric matrix and

$$Q={𝐗}^{\mathrm{T}}\mathrm{𝐁𝐗}.$$

Again, in the example of $X^2-5XY$, over $\mathbb{Q}$ it can be written as

However, it is not possible to represent $X^2-5XY$ over $\mathbb{Z}$ 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 $N=R[X_1,\mathrm{\dots },X_{n-1}]$. Let $\varphi :M\to N$ be the (unique) evaluation homomorphism of the embedding, with $\varphi (X_i)=X_i$ for and $\varphi (X_n)=0$. Then for any quadratic form $Q\in M$, $\varphi (Q)$ is a quadratic form in $N$.

In particular, if we take $N=R$, and $𝐬=(s_1,\mathrm{\dots },s_n)$ with $s_i\in R$. Then the evaluation homomorphism $\varphi$ at $𝐬$ for any quadratic form $Q\in M$ is called the evaluation of $Q$ at $𝐬$, and we write it ${\varphi }_{𝐬}(Q)$, or simply $Q(𝐬)$ (since $\varphi$ is uniquely determined by $𝐬$). In this way, a quadratic form $Q$ can be realized as a quadratic map, as follows:

Let $Q\in M$ be a qudratic form. Take the direct sum of $n$ copies of $R$ and call this $V$. Define a map $q:V\to R$ by $q(v)=Q(v)$. Then $q$ is a quadratic map.

Conversely, if $2$ is invertible in $R$ (so that $\mathrm{char}(R)\ne 2$ is clear), then given a quadratic map $q:M\to R$, one can find a corresponding quadratic form $Q\in M$ such that $q(v)=Q(v)$, by setting

$$a_{ij}=\frac{1}{2}\left(q(e_i+e_j)-q(e_i)-q(e_j)\right),$$

where $e_i$ and $e_j$ are coordinate vectors whose coordinates are all $0$ except at positions $i$ and $j$ respectively, where the coordinates are $1$. Then $Q$ defined by ${𝐗}^{\mathrm{T}}\mathrm{𝐀𝐗}$, where $𝐀=\{a_{ij}\}$ 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 $Q_1$ and $Q_2$ are said to be if there is an invertible matrix $M$ such that $Q_1(v)=Q_2(Mv)$, for all $v\in R^n$. 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 ${𝐀}_1$ and ${𝐀}_2$ are matrices corresponding to (see the definition section) the two equivalent quadratic forms $Q_1$ and $Q_2$ above, then ${𝐀}_1=M^{\mathrm{T}}{𝐀}_{2}M$.

For example, the quadratic form $X^2-Y^2$ is equivalent to $XY$ over any ring $R$ where $2$ is invertible, with .

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 $Q_1$ and $Q_2$ are two quadratic forms in $m$ and $n$ indeterminates. We can define a quadratic form $Q$ in $m+n$ indeterminates in terms of $Q_1$ and $Q_2$, called the sum of $Q_{\mathrm{1}}$ and $Q_{\mathrm{2}}$, as follows:

write $Q_1={𝐗}^{\mathrm{T}}\mathrm{𝐀𝐗}$ and $Q_2={𝐘}^{\mathrm{T}}\mathrm{𝐁𝐘}$, with $𝐗={(X_1,\mathrm{\dots },X_m)}^{\mathrm{T}}$ and $𝐘={(Y_1,\mathrm{\dots },Y_n)}^{\mathrm{T}}$. Then

$$Q:={𝐙}^{\mathrm{T}}(𝐀\oplus 𝐁)𝐙,$$

where $𝐙=(𝐗,𝐘)={(X_1,\mathrm{\dots },X_m,Y_1,\mathrm{\dots },Y_n)}^{\mathrm{T}}$, and $𝐀\oplus 𝐁$ is the direct sum of matrices $𝐀$ and $𝐁$.

Expressed in terms of $Q_1$ and $Q_2$, we write $Q=Q_1\oplus Q_2$. For example, if $Q_1=5X_1^2+6X_2^2$ and $Q_2=10X_1{X}_2$, then

$$Q_1\oplus Q_2=5X_1^2+6X_2^2+10X_3{X}_4,$$

not $5X_1^2+6X_2^2+10X_1{X}_2(=Q_1+Q_2)$.

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