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quadratic form (Definition)

In this entry, unless otherwise specified, $ R$ is a commutative ring with multiplicative identity $ 1$ and $ M=R[X_1,\ldots,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

$\displaystyle Q=a_{11}X_1^2+a_{12}X_1X_2+\cdots+a_{n,n-1}X_nX_{n-1}+a_{nn}X_n^2=\sum_{1\le i,j \le n}a_{ij}X_iX_j$
where $ a_{ij}\in R$.

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

$\displaystyle Q={\mathbf{X}}^{\mathrm{T}} \mathbf{A}\mathbf{X}.$

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

$\displaystyle X^2-5XY=\begin{pmatrix}X & Y \end{pmatrix}\begin{pmatrix}1 & -2 \\ -3 & 0 \end{pmatrix}\begin{pmatrix}X \\ Y \end{pmatrix}.$

Now suppose the characteristic of $ R$, $ \operatorname{char}(R)\ne 2$. In fact, suppose that $ 2$ is invertible in $ R$ (its inverse denoted by $ \frac{1}{2}$). Since $ X_iX_j=X_jX_i$, define $ b_{ij}=\frac{1}{2}(a_{ij}+a_{ji})$. Then $ b_{ii}=a_{ii}$ and $ b_{ij}=b_{ji}$. Furthermore, if $ \mathbf{B}=\lbrace b_{ij}\rbrace$, then $ \mathbf{B}$ is a symmetric matrix and

$\displaystyle Q={\mathbf{X}}^{\mathrm{T}} \mathbf{B}\mathbf{X}.$

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

$\displaystyle X^2-5XY=\begin{pmatrix}X & Y \end{pmatrix}\begin{pmatrix}1 & -\frac{5}{2} \\ -\frac{5}{2} & 0 \end{pmatrix}\begin{pmatrix}X \\ Y \end{pmatrix}.$
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,\ldots,X_{n-1}]$. Let $ \phi:M\to N$ be the (unique) evaluation homomorphism of the embedding, with $ \phi(X_i)=X_i$ for $ i<n$ and $ \phi(X_n)=0$. Then for any quadratic form $ Q\in M$, $ \phi(Q)$ is a quadratic form in $ N$.

In particular, if we take $ N=R$, and $ \mathbf{s}=(s_1,\ldots,s_n)$ with $ s_i\in R$. Then the evaluation homomorphism $ \phi$ at $ \mathbf{s}$ for any quadratic form $ Q\in M$ is called the evaluation of $ Q$ at $ \mathbf{s}$, and we write it $ \phi_{\mathbf{s}}(Q)$, or simply $ Q(\mathbf{s})$ (since $ \phi$ is uniquely determined by $ \mathbf{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 $ \operatorname{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

$\displaystyle a_{ij}=\frac{1}{2}\big(q(e_i+e_j)-q(e_i)-q(e_j)\big),$
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 $ {\mathbf{X}}^{\mathrm{T}} \mathbf{A}\mathbf{X}$, where $ \mathbf{A}=\lbrace a_{ij}\rbrace$ 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 equivalent if there is a non-singular 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 $ \mathbf{A}_1$ and $ \mathbf{A}_2$ are matrices corresponding to (see the definition section) the two equivalent quadratic forms $ Q_1$ and $ Q_2$ above, then $ \mathbf{A}_1=M^\mathrm{T}\mathbf{A}_2M$.

For example, the quadratic form $ X^2-Y^2$ is equivalent to $ XY$ over any ring $ R$ where $ 2$ is invertible, with $ M=\begin{pmatrix}1 & -1 \\ 1 & 1 \end{pmatrix}$.

In the case where $ R=\mathbb{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_1$ and $ Q_2$, as follows:

write $ Q_1={\mathbf{X}}^{\mathrm{T}} \mathbf{A}\mathbf{X}$ and $ Q_2={\mathbf{Y}}^{\mathrm{T}} \mathbf{B}\mathbf{Y}$, with $ \mathbf{X}=(X_1,\ldots,X_m)^{\mathrm{T}}$ and $ \mathbf{Y}=(Y_1,\ldots,Y_n)^{\mathrm{T}}$. Then
$\displaystyle Q:={\mathbf{Z}}^{\mathrm{T}} (\mathbf{A}\oplus \mathbf{B}) \mathbf{Z},$
where $ \mathbf{Z}=(\mathbf{X},\mathbf{Y})=(X_1,\ldots,X_m,Y_1,\ldots,Y_n)^{\mathrm{T}}$, and $ \mathbf{A}\oplus \mathbf{B}$ is the direct sum of matrices $ \mathbf{A}$ and $ \mathbf{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_1X_2$, then
$\displaystyle Q_1\oplus Q_2=5X_1^2+6X_2^2+10X_3X_4,$
not $ 5X_1^2+6X_2^2+10X_1X_2 (=Q_1+Q_2)$.

Bibliography

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



"quadratic form" is owned by CWoo. [ full author list (4) | owner history (2) ]
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See Also: positive definite, negative definite, symmetric bilinear form, quadratic space, proof of Gaussian maximizes entropy for given covariance, isotropic quadratic space

Also defines:  equivalent quadratic forms, sum of quadratic forms, evaluation of a quadratic form

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quadratic map (Derivation) by Algeboy
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Cross-references: direct sum of matrices, sum, terms, positive semidefinite, positive definite, formally real field, real numbers, field, equivalent, section, equivalence relation, equivalence, well-defined, non-singular, vectors, coordinate, clear, map, direct sum, quadratic map, embedding, evaluation homomorphism, represent, symmetric matrix, inverse, invertible, characteristic, matrix, ring, degree, homogeneous polynomial, indeterminates, polynomial ring, multiplicative identity, commutative ring
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This is version 41 of quadratic form, born on 2002-02-13, modified 2008-02-12.
Object id is 1940, canonical name is QuadraticForm.
Accessed 23281 times total.

Classification:
AMS MSC: 15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
 11E04 (Number theory :: Forms and linear algebraic groups :: Quadratic forms over general fields)
 11E08 (Number theory :: Forms and linear algebraic groups :: Quadratic forms over local rings and fields)
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