Consider the following symmetric tridiagonal block matrix: $$\begin{bmatrix} 2I_{N \times N} & -I_{N \times N} & O & \dots & O &O \\ -I_{N \times N} & 2I_{N \times N} & -I_{N \times N} & \dots & O&O \\ O & -I_{N \times N} & 2I_{N \times N} & -I_{N \times N} \dots & O&O \\ \vdots & \vdots & \vdots & \ddots & \vdots &O \\ \vdots & \vdots & \vdots & \ddots & \vdots &-I_{N \times N} \\ O & O & O & \dots &-I_{N \times N} & 2I_{N \times N} \end{bmatrix}_{N^2 \times N^2}$$
Above matrix comes from an intermediate step of higher-dimensional 2nd order finite difference scheme of Poisson equation. I'm wondering what are the eigenvalues of above matrix. What I know currently is for following (SPD) tridiagonal matrix $$\begin{bmatrix} 2 & -1 & 0 & \dots & 0 &0\\ -1 & 2 & -1 & \dots & 0&0 \\ 0 & -1 & 2 & -1& \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots&0 \\ \vdots & \vdots & \vdots & \ddots & \vdots&-1 \\ 0 & 0 & 0 & \dots &-1 & 2 \end{bmatrix}_{N \times N}$$ It has eigenvalues $\lambda_j = 4sin^2(\frac{j \pi}{2(N+1)})$, $1 \leq j \leq N$.
Any help would be appreciated!