Finding the matrix of this particular quadratic form I have been working on problems related to bilinear and quadratic forms,
and I came across an introductory problem that I have been having issues with.
Take
$$Q(x) = x_1^2 + 2x_1x_2 - 3x_1x_3 - 9x_2^2 + 6x_2x_3 + 13x_3^2$$
I want to find a matrix $A$ such that $Q(x) = \langle Ax,x \rangle$. My initial guess was to simply establish this via a coefficient matrix, i.e.,
$$A = \begin{bmatrix} 1 & 2 & -3\\ 2 & -9 & 6\\ -3 & 6 & 13\end{bmatrix}$$
However, upon closer inspection, I see that this matrix does not produce our
desired outcome. Is there a more reasonable algorithm for generating the
matrix $A$ of a quadratic form?

The matrix associated with your quadratic form is
$$Q=\begin{pmatrix}1 & 1 & -\frac{3}{2} \\ 1 & -9 & 3 \\ -\frac{3}{2} & 3 & 13 \end{pmatrix}$$
and the characteristic polynomial of $Q$ is 
$$ q(x) = x^3-5x^2-\frac{501}{4}x+\frac{511}{4}$$
so the eigenvalues are 
$$\approx -9.5372,\qquad \approx 0.9886,\qquad \approx 13.5486. $$
