I'm following a course on numerical methods and we're discussing the finite-difference method in $2\text{D.}$
This question is about the proof that the system matrix is symmetric with dirichlet boundary's.
The proof on the slides goes as follows:
$\begin{aligned} \boldsymbol{v}^{T} \boldsymbol{A} \boldsymbol{u}=& \sum_{j=1}^{N_{y}-1} \sum_{i=1}^{N_{x}-1} v_{i, j} \frac{-u_{i-1, j}-u_{i, j-1}+4 u_{i, j}-u_{i, j+1}-u_{i+1, j}}{h^{2}} \\=& \sum_{j=1}^{N_{y}-1} \sum_{i=1}^{N_{x}-1} v_{i, j}\left(\frac{u_{i, j}-u_{i-1, j}}{h^{2}}-\frac{u_{i+1, j}-u_{i, j}}{h^{2}}\right.\\+\frac{u_{i, j}-u_{i, j-1}}{h^{2}}-\frac{u_{i, j+1}-u_{i, j}}{h^{2}} &=\end{aligned}$
For any fixed $j = 1,.....N_{y-1}$
\begin{equation}\begin{array}{l} \sum_{i=1}^{N_{x}-1} v_{i, j}\left(\frac{u_{i, j}-u_{i-1, j}}{h^{2}}-\frac{u_{i+1, j}-u_{i, j}}{h^{2}}\right) \\ =\sum_{i=1}^{N_{x}-1} v_{i, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}}-\sum_{i=1}^{N_{x}-1} v_{i, j} \frac{u_{i+1, j}-u_{i, j}}{h^{2}} \\ =\sum_{i=1}^{N_{x}-1} v_{i, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}}-\sum_{i=2}^{N_{x}} v_{i-1, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}} \\ =\sum_{i=1}^{N_{x}} v_{i, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}}-\sum_{i=1}^{N_{x}} v_{i-1, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}} \\ =\sum_{i=1}^{N_{x}} \frac{v_{i, j}-v_{i-1, j}}{h} \frac{u_{i, j}-u_{i-1, j}}{h} \end{array}\end{equation}
I'm confused about this step:
$\begin{aligned} &=\sum_{i=1}^{N_{x}-1} v_{i, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}}-\sum_{i=2}^{N_{x}} v_{i-1, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}} \\ &=\sum_{i=1}^{N_{x}} v_{i, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}}-\sum_{i=1}^{N_{x}} v_{i-1, j} \frac{u_{i, j}-u_{i-1, j}}{h^{2}} \end{aligned}$
One step before the confusing part they change the summation interval from $[i=1,...,N_{x-1}]$ to $[i=2,...,N_{x}]$ of the most right term. They also changed the indices of the variables, so I see how it is justified.
In the confusing part they change the interval from $[i=1,...,N_{x-1}]$ to $[i=1,...,N_{x}]$ of the most left side, but without changing the indices. So they just do an extra summation. Honestly I don't see how that is justified.
Could I get some feedback about that?
Ter