Quadratic Forms

The general expression of a quadratic form in \(3\) variables is given below:

\[ \begin{aligned} f(x, y, z) &= \mathbf{x}^T \mathbf{A} \mathbf{x}\\\\ &= a_{xx} x^2 + a_{yy}y^2 + a_{zz} z^2 + (a_{xy} + a_{yx}) xy + (a_{yz} + a_{zy}) yz + (a_{zx} + a_{xz}) zx \end{aligned} \]

Here \(\mathbf{x} = \begin{bmatrix}x\\y\\z\end{bmatrix}\) and \(\mathbf{A} =\begin{bmatrix}a_{xx} & a_{xy} & a_{xz}\\a_{yx} & a_{yy} & a_{yz}\\a_{zx} & a_{zy} & a_{zz}\end{bmatrix}\)