I'm trying to discretize the Laplacian operator, and represent it with a matrix, but I'm running into a problem: my result is not hermitian when it should be. Here are my calculations:
In one dimension...
$\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}}$
Deriving the discrete representation of the Laplacian...
Forward Difference: $\frac{\partial}{\partial x} \rightarrow \frac{F[i+1]- F[i]}{\Delta}$
Reverse Difference: $\frac{\partial}{\partial x} \rightarrow \frac{F[i]- F[i-1]}{\Delta}$
Taking the derivative of the forward difference via the reverse difference definition:
$\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} \rightarrow \frac{F[i+1]-F[i]}{\Delta^{2}} - \frac{F[i]-F[i-1]}{\Delta^{2}} = \frac{F[i+1]-2F[i]+F[i-1]}{\Delta^{2}}$
Great, so let's put this in matrix form for N = 4 (indexed from 1 and omitting $\Delta^{2}$):
$\nabla^{2} \rightarrow \begin{matrix} -2F[1] & F[2] & 0 & 0 \\ F[1] & -2F[2] & F[3] & 0 \\ 0 & F[2] & -2F[3] & F[4] \\ 0 & 0 & F[3] & -2F[4] \end{matrix}$
It is quite clear that that matrix is not hermitian. The indices do not match after transposition. What am I missing?