A quadratic form is a homogeneous polynomial of degree two in a number of variables. An $n$-ary quadratic form over a field K is a homogeneous polynomial of degree 2 in $n$ variables with coefficients in $K$ :
$q(x_1 , ..., x_n)=\sum_{i,j=1}^{n} a_{ij} x_i x_j$ ,     $a_{ij} \in K$.
Let $\textbf{x}$ be the column vector with components $x_1$, ..., $x_n$ and $A = (a_{ij})$ be the $n \times n$ matrix over $K$ whose entries are the coefficients of $q$.
Then, $q(x) = \textbf{x}^T A \textbf{x}$.