Editing Quadratic form (section)

=== General case ===
Given a quadratic form {{math|''q''<sub>''A''</sub>}} over the real numbers, defined by the matrix {{math|1=''A'' = (''a''<sub>''ij''</sub>)}},
the matrix
<math display = block>B = \left(\frac{a_{ij}+a_{ji}} 2\right) = \frac{1} 2(A + A^\text{T})</math>
is [[symmetric matrix|symmetric]], defines the same quadratic form as {{mvar|A}}, and is the unique symmetric matrix that defines {{math|''q''<sub>''A''</sub>}}.

So, over the real numbers (and, more generally, over a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] different from two), there is a [[one-to-one correspondence]] between quadratic forms and [[symmetric matrices]] that determine them.