Quadratic Form¶

Definition¶

A quadratic form on $\mathbb{R}^n$ is a function $Q\colon\mathbb{R}^n\to\mathbb{R}$ of the form

Q(\boldsymbol{v})=\boldsymbol{v}^{\top}\boldsymbol{A}\boldsymbol{v}=\sum_{i=1}^n\sum_{j=1}^n a_{ij}v_iv_j,

where $\boldsymbol{A}=(a_{ij})$ is a real $n\times n$ matrix and $\boldsymbol{v}$ a column vector. Equivalently, $Q$ is a polynomial in $v_1,\ldots,v_n$ whose terms all have degree exactly two (a homogeneous polynomial of degree two, the zero polynomial included, which comes from $\boldsymbol{A}=\boldsymbol{0}$ $0$ ). The matrix can always be chosen symmetric, and the symmetric choice is unique: it is called the matrix of $Q$ .

Interpretation¶

A quadratic form is the many-variable version of $ax^2$ : every term has degree exactly two, so $Q(c\boldsymbol{v})=c^2Q(\boldsymbol{v})$ $c^{2} Q (v)$ , and a symmetric matrix stores the coefficients. The main question about a quadratic form is its sign in different directions.

Two panels of level curves: nested ellipses of x1 squared plus 2 x2 squared, positive definite, and hyperbolas of x1 squared minus x2 squared with asymptotes, indefinite.

The figure shows curves on which the form is constant. For $x_1^2+2x_2^2$ $2 x_{2}^{2}$ , with matrix $\operatorname{diag}(1,2)$ , they are nested ellipses: the value is positive in every direction. For $x_1^2-x_2^2$ $x_{2}^{2}$ , with matrix $\operatorname{diag}(1,-1)$ , they are hyperbolas: the form is positive near the $x_1$ -axis, negative near the $x_2$ -axis and zero on the dashed lines $x_2=\pm x_1$ $\pm x_{1}$ . Because $Q(c\boldsymbol{v})=c^2Q(\boldsymbol{v})$ $c^{2} Q (v)$ , the sign is the same along each line through the origin, apart from the origin itself.

The transpose turns the column vector $\boldsymbol{v}$ into a row, so the matrix products in $\boldsymbol{v}^{\top}\boldsymbol{A}\boldsymbol{v}$ have sizes $1\times n$ $n$ , $n\times n$ and $n\times1$ and produce a number. For $n=2$ and a symmetric $\boldsymbol{A}=\begin{pmatrix}a&b\\b&d\end{pmatrix}$ $(a b b d)$ , expanding gives $av_1^2+2bv_1v_2+dv_2^2$ $d v_{2}^{2}$ : the diagonal entries are the coefficients of the squares, and each off-diagonal entry carries half of a mixed term.

Sorting forms by their signs gives the definiteness of the matrix. Quadratic forms return in calculus of several variables, where the second-order term of a Taylor approximation is a quadratic form in the displacement.

A common slip is to put the whole coefficient of a mixed term into the matrix. The form $3x^2+4xy+5y^2$ has the symmetric matrix $\begin{pmatrix}3&2\\2&5\end{pmatrix}$ ; the matrix $\begin{pmatrix}3&4\\4&5\end{pmatrix}$ would give $8xy$ .

Key Results¶

Different matrices can give the same form: $\boldsymbol{v}^{\top}\boldsymbol{A}\boldsymbol{v}=\boldsymbol{v}^{\top}\bar{\boldsymbol{A}}\boldsymbol{v}$ for $\bar{\boldsymbol{A}}=\tfrac{1}{2}(\boldsymbol{A}+\boldsymbol{A}^{\top})$ $A^{⊤})$ , because $\boldsymbol{v}^{\top}\boldsymbol{A}\boldsymbol{v}=\boldsymbol{v}^{\top}\boldsymbol{A}^{\top}\boldsymbol{v}$ $v^{⊤} A^{⊤} v$ . The symmetric matrix $\bar{\boldsymbol{A}}$ is the only symmetric one with this form.
In terms of the symmetric matrix, the coefficient of $v_i^2$ is $a_{ii}$ and the coefficient of $v_iv_j$ ( $i \neq =$ $i\ne j$ $j$ ) is $2a_{ij}$ .
Quadratic forms are used to define definiteness of symmetric matrices.

Examples¶

If $\boldsymbol{A}=\boldsymbol{I}_n$ $I_{n}$ , then $\boldsymbol{v}^{\top}\boldsymbol{A}\boldsymbol{v}=\|\boldsymbol{v}\|^2$ $∥ v ∥^{2}$ .
For $\boldsymbol{A}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ $(10 0 - 1)$ , $\boldsymbol{v}^{\top}\boldsymbol{A}\boldsymbol{v}=x^2-y^2$ $y^{2}$ .
$Q(x,y)=3x^2+4xy+5y^2$ has the symmetric matrix $\begin{pmatrix}3&2\\2&5\end{pmatrix}$ ; the non-symmetric $\begin{pmatrix}3&4\\0&5\end{pmatrix}$ gives the same form.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 6
Math III: Analysis · Lecture 7

Aliases: homogeneous quadratic polynomial Symbols: real numbers, sum

Last updated: 27 October 2026 11:52