We define the matrix for the discrete Poisson problem as
\begin{equation*} \begin{bmatrix} T & -I & 0 & \cdots & \cdots & \cdots & 0 \\ -I & T & -I & \ddots & & & \vdots \\ 0 & \ddots & \ddots & \ddots & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & \ddots &\ddots & -I \\ 0 & \cdots & \cdots & \cdots & 0 & -I & T \\ \end{bmatrix} \in \mathbb R^{N^2\times N^2} \end{equation*}
with $I \in \mathbb R^{N\times N}$ being the identity matrix and $T\in \mathbb R^{N\times N}$ being a tridiagonal matrix with $4$ at main diagonal and $-1$ at the upper and sub diagonal.
So my task is to find the eigenvalues and -vectors of this matrix. I had no idea at first, so I just tried calculating them for the first few N, to see, if I could find a pattern. Unfortunately, that didn't help me. But I don't know any other way nor do I have an idea how to find those values. I'd be glad, if someone could help me here.