Quadratic form (이차형식)



Quadratic form
A quadratic form is a homogeneous polynomial of degree two in a number of variables. An $$$$n$$$$-ary quadratic form over a field K is a homogeneous polynomial of degree 2 in $$$$n$$$$ variables with coefficients in $$$$K$$$$ :
$$$$q(x_1 , ..., x_n)=\sum_{i,j=1}^{n} a_{ij} x_i x_j$$$$ ,     $$$$a_{ij} \in K $$$$.

This formula may be rewritten using matrices :
Let $$$$\textbf{x}$$$$ be the column vector with components $$$$x_1$$$$, ..., $$$$x_n$$$$ and $$$$A = (a_{ij})$$$$ be the $$$$n \times n$$$$ matrix over $$$$K$$$$ whose entries are the coefficients of $$$$q$$$$.
Then, $$$$q(x) = \textbf{x}^T A \textbf{x}$$$$.