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}@@.