Is there a systematic way of finding the matrix of a quadratic form?

Question

For example i have this quadratic form $q(x_1,x_2)=8{x_1}^2-4x_1x_2+5{x_2}^2$ , here it's a simple factoring:

$q\begin{bmatrix}x_1 \\x_2 \\x_3\\\end{bmatrix}=\begin{bmatrix}x_1 \\x_2 \\x_3\\\end{bmatrix} \cdot \begin{bmatrix}8x_1 &-2x_2\\-2x_1&5x_2\end{bmatrix}=\vec{x}^{T}A\vec{x} ,A=\begin{bmatrix}8 &-2\\-2&5\end{bmatrix}$.

But this is not always the case where one can simply see how the matrix is going to be ,so is there a certain method of finding this matrix?

Answer 1

The matrix of the quadratic form $q(x_1,x_2)=a{x_1}^2+bx_1x_2+c{x_2}^2$ is always$$\begin{pmatrix}a&\frac b2\\\frac b2&c\end{pmatrix}.$$

Answer 2

The general method is to write the coefficients of the quadratic terms on the diagonal. And for the $a_{ij}x_ix_j$ terms write $a_{ij}/2$ into the i.th row and j.th column and also into the j.th row and the i.th column.