Symmetry in the quadratic form matrix The following is an excerpt from Greene's Econometric Analysis (7th Edition). Therein, the author states the matrix in quadratic forms must be symmetric. I would like to know why. What if, for instance, $a_{12} \neq a_{21}$?


$$\begin{pmatrix}x&y\end{pmatrix}\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=ax^2+(b+c)xy+dy^2$$ can be symmetrized as
$$\begin{pmatrix}x&y\end{pmatrix}\begin{pmatrix}a&\frac{b+c}2\\\frac{b+c}2&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=ax^2+2\frac{b+c}2xy+dy^2.$$
So a quadratic form has a symmetric matrix WLOG.
