# 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 $\mathbf{X}={({X}_{1},\mathrm{\dots},{X}_{n})}^{\mathrm{T}}$, and $\mathbf{A}=\{{a}_{ij}\}$ the $n\times n$ matrix, then we can rewrite $Q$ as

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

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

$${X}^{2}-5XY=\left(\begin{array}{cc}\hfill X\hfill & \hfill Y\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 1\hfill & \hfill -2\hfill \\ \hfill -3\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{c}\hfill X\hfill \\ \hfill Y\hfill \end{array}\right).$$ |

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 $\mathbf{B}=\{{b}_{ij}\}$, then $\mathbf{B}$ is a symmetric matrix^{} and

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

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

$${X}^{2}-5XY=\left(\begin{array}{cc}\hfill X\hfill & \hfill Y\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 1\hfill & \hfill -\frac{5}{2}\hfill \\ \hfill -\frac{5}{2}\hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{c}\hfill X\hfill \\ \hfill Y\hfill \end{array}\right).$$ |

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 $\mathbf{s}=({s}_{1},\mathrm{\dots},{s}_{n})$ with ${s}_{i}\in R$. Then the evaluation homomorphism $\varphi $ at $\mathbf{s}$ for any quadratic form $Q\in M$ is called the *evaluation* of $Q$ at $\mathbf{s}$, and we write it ${\varphi}_{\mathbf{s}}(Q)$, or simply $Q(\mathbf{s})$ (since $\varphi $ 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 $\mathrm{char}(R)\ne 2$ 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 ${\mathbf{X}}^{\mathrm{T}}\mathrm{\mathbf{A}\mathbf{X}}$, where $\mathbf{A}=\{{a}_{ij}\}$ 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 ${\mathbf{A}}_{1}$ and ${\mathbf{A}}_{2}$ are matrices corresponding to (see the definition section^{}) the two equivalent quadratic forms ${Q}_{1}$ and ${Q}_{2}$ above, then ${\mathbf{A}}_{1}={M}^{\mathrm{T}}{\mathbf{A}}_{2}M$.

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

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 ${Q}_{\mathrm{1}}$ and ${Q}_{\mathrm{2}}$*, as follows:

write ${Q}_{1}={\mathbf{X}}^{\mathrm{T}}\mathrm{\mathbf{A}\mathbf{X}}$ and ${Q}_{2}={\mathbf{Y}}^{\mathrm{T}}\mathrm{\mathbf{B}\mathbf{Y}}$, with $\mathbf{X}={({X}_{1},\mathrm{\dots},{X}_{m})}^{\mathrm{T}}$ and $\mathbf{Y}={({Y}_{1},\mathrm{\dots},{Y}_{n})}^{\mathrm{T}}$. Then

$$Q:={\mathbf{Z}}^{\mathrm{T}}(\mathbf{A}\oplus \mathbf{B})\mathbf{Z},$$ where $\mathbf{Z}=(\mathbf{X},\mathbf{Y})={({X}_{1},\mathrm{\dots},{X}_{m},{Y}_{1},\mathrm{\dots},{Y}_{n})}^{\mathrm{T}}$, 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 ${Q}_{1}=5{X}_{1}^{2}+6{X}_{2}^{2}$ and ${Q}_{2}=10{X}_{1}{X}_{2}$, then

$${Q}_{1}\oplus {Q}_{2}=5{X}_{1}^{2}+6{X}_{2}^{2}+10{X}_{3}{X}_{4},$$ |

not $5{X}_{1}^{2}+6{X}_{2}^{2}+10{X}_{1}{X}_{2}\phantom{\rule{veryverythickmathspace}{0ex}}(={Q}_{1}+{Q}_{2})$.

## References

- 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 |