Hi. I am trying to determine if $L=-8\partial_1^2+4\partial_1\partial_2-\partial^2_2+3\partial_2$ in $\mathbb{R^2}$ is elliptic.
The easiest way I can think of doing this is to find the eigenvalues of the symmetric matrix that corresponds to $L$. I know that if we have an operator of the form $$P=a\partial_1^2+b\partial_2^2\implies A=\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}.$$ But I do not know how to form a symetric matrix for $L$. Thank you kindly.