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 $5X_1^2+6X_2^2+10X_1{X}_2(=Q_1+Q_2)$.