It's said that if I have the elliptic partial differential equations: $\sum_{i,j=1}^n a_{ij}\frac {\partial ^2 u}{\partial x_i^2 \partial x_j^2}$, it's transferable to $\nabla ^2 = \frac {\partial ^2}{\partial x_i ^2}$ by the linear Laplace transform iff the matrix of this transformation $A$ is positive semi-definitive, but why the positive semi-definiteness?
I was trying to find some proof but it was too hard for me, I just want to understand it.