Editing Quadratic form (section)

=== Example ===
Consider the case of quadratic forms in three variables {{math|''x'', ''y'', ''z''}}. The matrix {{mvar|A}} has the form
<math display="block">A=\begin{bmatrix}
  a&b&c\\d&e&f\\g&h&k
\end{bmatrix}.</math>

The above formula gives
<math display="block">q_A(x,y,z)=ax^2 + ey^2 +kz^2 + (b+d)xy + (c+g)xz + (f+h)yz.</math>

So, two different matrices define the same quadratic form if and only if they have the same elements on the diagonal and the same values for the sums {{math|''b'' + ''d''}}, {{math|''c'' + ''g''}} and {{math|''f'' + ''h''}}. In particular, the quadratic form {{math|''q''<sub>''A''</sub>}} is defined by a unique [[symmetric matrix]]
<math display="block">A=\begin{bmatrix}
  a&\frac{b+d}2&\frac{c+g}2\\
  \frac{b+d}2&e&\frac{f+h}2\\
  \frac{c+g}2&\frac{f+h}2&k
\end{bmatrix}.</math>

This generalizes to any number of variables as follows.