Quadratic forms are the degree-two part of algebra. They appear when a function is approximated to second order, when a symmetric matrix is studied through its associated scalar expression, and when a positive definite form supplies the square of a length. General quadratic forms are broader than metric geometry: they may be indefinite or degenerate, and their zero sets can contain nonzero vectors. In linear algebra they translate symmetric matrices into geometry; in algebra they encode bilinear data with arithmetic content; in analysis they underlie energy functionals and elliptic estimates. The same object connects [bilinear forms](/page/Bilinear%20Form), [inner product spaces](/page/Inner%20Product%20Space), [symmetric matrices](/page/Symmetric%20Matrix), [eigenvalues](/page/Eigenvalue), and [quadratic polynomials](/page/Polynomial).

A first way to meet a quadratic form is through real coordinates. If $A$ is a symmetric matrix, then $x \mapsto x^\top A x$ is a scalar quantity built from all pairwise products $x_i x_j$. Symmetry matters because the expression only sees the symmetric part of a matrix: the antisymmetric part vanishes when placed between $x^\top$ and $x$.

## Definition

The core notion should not depend on a preferred coordinate system. A quadratic form is the kind of scalar-valued function whose value scales like a square and whose mixed second-order part is bilinear. Over fields of characteristic not equal to $2$, this gives a clean intrinsic definition and keeps the familiar matrix formula as a coordinate representation rather than the definition itself.

[definition: Quadratic Form] Let $F$ be a field with $\operatorname{char}(F) \ne 2$, and let $V$ be a [vector space](/page/Vector%20Space) over $F$. A function $q: V \to F$ is a quadratic form if $q(av)=a^2q(v)$ for all $a \in F$ and $v \in V$, and the function $B_q: V \times V \to F$ defined by \begin{align*} B_q(u,v)=\frac{1}{2}\bigl(q(u+v)-q(u)-q(v)\bigr) \end{align*} is a symmetric bilinear form. [/definition]

This definition makes the parent concept independent of coordinates. It also names the associated symmetric bilinear form $B_q$, which is the object used to define radicals, nondegeneracy, and matrix representatives later on.

The most concrete way to compute with the definition is to choose real coordinates. This is the form that appears in multivariable calculus, optimization, spectral theory, and the classification of conic sections. It packages a homogeneous degree-two polynomial in $n$ real variables as a matrix expression.

[definition: Real Quadratic Form] A real quadratic form on $\mathbb{R}^n$ is a function $Q: \mathbb{R}^n \to \mathbb{R}$ for which there exists a real symmetric matrix $A \in \mathbb{R}^{n \times n}$ such that \begin{align*} Q(x) = x^\top A x \end{align*} for every $x \in \mathbb{R}^n$. [/definition]

Writing $x = (x_1, \ldots, x_n)$ reveals the polynomial hidden inside the matrix formula:

\begin{align*} x^\top A x = \sum_{i=1}^n \sum_{j=1}^n A_{ij}x_i x_j. \end{align*}

If $A$ is symmetric, the off-diagonal terms occur in matched pairs $A_{ij}x_i x_j$ and $A_{ji}x_jx_i$.

Matrix formulas are excellent for calculation, but they hide the invariant source of many quadratic forms. In geometry and functional analysis, the primary object is often a bilinear pairing that measures how two vectors interact; energies, squared lengths, and second variations are then obtained by feeding the same vector into both slots. To compare that two-vector data with a one-vector quadratic quantity, we need a named construction that records precisely the diagonal part of a bilinear form.

[definition: Quadratic Form Associated to a Bilinear Form] Let $F$ be a field, let $V$ be a vector space over $F$, and let $B: V \times V \to F$ be a bilinear form. The quadratic form associated to $B$ is the function $q_B: V \to F$ defined by \begin{align*} q_B(v) = B(v,v). \end{align*} [/definition]

This construction separates two pieces of structure: the bilinear form remembers interactions between two possibly different vectors, while the quadratic form records only the diagonal values. Over fields of characteristic not equal to $2$, the diagonal function depends only on the symmetric part of $B$, and symmetric bilinear forms can be recovered from their quadratic forms by polarization. In characteristic $2$, this relationship changes in a serious way, so the distinction is not cosmetic.

Many algebraic questions require allowing coefficients in rings rather than fields. For example, integral quadratic forms remember divisibility and congruence information that disappears after extending scalars to $\mathbb{Q}$ or $\mathbb{R}$. The polynomial definition is the most stable way to state the concept over a commutative ring.

[definition: Quadratic Form Over a Commutative Ring] Let $R$ be a commutative ring. A quadratic form in $n$ variables over $R$ is a map $q: R^n \to R$ together with a homogeneous coordinate expression \begin{align*} q(x_1, \ldots, x_n)=\sum_{1 \le i \le j \le n} a_{ij}x_i x_j \end{align*} for some coefficients $a_{ij} \in R$, such that $q(x_1,\ldots,x_n)$ is evaluated by this expression for every $(x_1,\ldots,x_n) \in R^n$. [/definition]

The ring definition keeps both pieces visible: the graph-context object is a function $q: R^n \to R$, while the homogeneous expression records the coefficients that make it a quadratic form. Over a general ring distinct polynomial expressions can define the same function on the underlying set, and not every operation involving division by $2$ is available. The coefficient expression is therefore part of the data used in algebraic and arithmetic questions, not merely a disposable way to evaluate the function.

Over rings, the same polynomial can carry arithmetic data that is invisible over the [real numbers](/page/Real%20Numbers). Integral quadratic forms are central in number theory because they ask which integers can be represented by a fixed [homogeneous polynomial](/page/Homogeneous%20Polynomial).

[example: Integral Binary Quadratic Form] Let $q: \mathbb{Z}^2 \to \mathbb{Z}$ be defined by \begin{align*} q(x,y)=x^2+xy+y^2. \end{align*} This is the function induced by a quadratic form over the commutative ring $\mathbb{Z}$, since the expression is homogeneous of degree $2$ with integer coefficients. It represents $1$ because \begin{align*} q(1,0)=1^2+1\cdot 0+0^2=1, \end{align*} and it represents $3$ because \begin{align*} q(1,1)=1^2+1\cdot 1+1^2=3. \end{align*} We show that $q$ does not represent $2$ over $\mathbb{Z}^2$. Suppose, for contradiction, that $x,y \in \mathbb{Z}$ satisfy $q(x,y)=2$. Then \begin{align*} x^2+xy+y^2=2. \end{align*} Multiplying by $4$ gives \begin{align*} 4x^2+4xy+4y^2=8. \end{align*} Since \begin{align*} (2x+y)^2=4x^2+4xy+y^2, \end{align*} we have \begin{align*} 4x^2+4xy+4y^2=(2x+y)^2+3y^2. \end{align*} Therefore \begin{align*} (2x+y)^2+3y^2=8. \end{align*} The term $(2x+y)^2$ is nonnegative, so $3y^2 \le 8$. Hence $y^2 \le 2$, and because $y \in \mathbb{Z}$ this forces $y \in \{-1,0,1\}$. If $y=0$, then \begin{align*} (2x+y)^2=8, \end{align*} which is impossible because $2x+y$ is an integer and no integer square equals $8$. If $y=1$ or $y=-1$, then $3y^2=3$, so \begin{align*} (2x+y)^2=8-3=5, \end{align*} which is impossible because no integer square equals $5$. Thus no pair $(x,y) \in \mathbb{Z}^2$ satisfies $q(x,y)=2$. This example illustrates the arithmetic question attached to an integral form: determine the set of values $q(x,y)$ as $(x,y)$ ranges through $\mathbb{Z}^2$. [/example]

When a vector space has a fixed ordered basis, a quadratic form becomes computable through its coordinate column. This motivates attaching a matrix to the form, because determinants, rank, eigenvalues, and change-of-basis formulas all require a matrix representative.