For $n \geq 4$, $A$ is $(n-1) \times (n-1)$ tridiagonal block matrix $$A = n^2 \begin{bmatrix}B & -I & 0 & \cdots & \\-I & B & -I & 0 & \\ 0 & -I & B & -I & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & -I & B & -I \\ \cdots & \cdots & 0 & -I & B \end{bmatrix}$$
where $B$ is again $(n-1) \times (n-1)$ tridiagonal matrix
$$B = \begin{bmatrix}4 & -1 & 0 & \cdots & \\-1 & 4 & -1 & 0 & \\ 0 & -1 & 4 & -1 & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & -1 & 4 & -1 \\ \cdots & \cdots & 0 & -1 & 4 \end{bmatrix}$$
(Hence the size of $A$ is $(n-1)^2 \times (n-1)^2$)
The eigenvalues of $A$ are given by $4n^2 (\sin^2(\mu \pi /(2n)) + \sin^2(\nu \pi /(2n)))$ for $\mu, \nu = 1, 2, \cdots, n-1$.
My professor gave a problem computing the corresponding eigenvectors of $A$ using numerical analysis, and I have to compare it to the exact closed form of eigenvectors which my professor gave.
However, using Matlab, I checked that my professor's exact closed form is wrong (multiplying $A$ to that vector $v$ does not give some kind of $kv$ - I guess maybe there was some typo in his paper).
So, does anyone know any reference about the closed form of eigenvectors of this matrix? Any helps will be appreciated. Thanks.