A quadratic function on a vector space
$V$
is a homogeneous function of degree 2.
If
$\mathbf{v} = \begin{pmatrix}x_1\\x_2\\ \vdots \\x_n\end{pmatrix}$
then
$f(\mathbf{v})$
$f(t \mathbf{v}) =t^2 f(\mathbf{v})$
.
Any quadratic form can be written in matrix form. If
$f(x_1, x_2) =x_1^2+3x_1x_2+4x^2_2$
then we can write it in quadratic form as
$f(x_1,x_2)= (x_1,x_2) \left( \begin{array}{cc} 1 & 1.5 \\ 1.5 & 4 \end{array} \right) \begin{pmatrix}x_1\\x_2\end{pmatrix}$
.
The matrix that expresses the quadratic form
$f(x_1,x_2,...,x_n) =\sum_{ij, \: i \leq j} a_{ij} x_i^2x_j^2$
in matrix form as above is
$\left( \begin{array}{cccc} a_{11} & \frac{a_{12}}{2} & \ldots & \frac{a_{1n}}{2} \\ \frac{a_{12}}{2} & a_{22} & \ldots & \frac{a_{2n}}{2} \\ \vdots & \ddots & \ddots & \vdots \\ \frac{a_{1n}}{2} & \frac{a_{2n}}{2} & \ldots & a_{nn} \end{array} \right)$