I have the following matrix arising when I tried to discretize the Green function， now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute value less than 1 for certain choices of $N$.

Any one has any ideas about how to find the analytical representation of eigenvalues of the matrix $G$, i,e, the eigenvalues represented by $N$? Thank you so much for any help!

$$G^{(s)} = \frac{h}{N + 1}\begin{pmatrix} - N & - N + 1 & \ldots & - 2 & - 1 \\ - N + 1 & 2( - N + 1) & \ddots & - 4 & - 2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ - 2 & - 4 & \ddots & ( - 1 + N)( - 2) & ( - 1 + N)( - 1) \\ - 1 & - 2 & \ldots & ( - 1 + N)( - 1) & N( - 1) \\ \end{pmatrix}$$