Consider a matrix $A \in M_n(\mathbb{R})$ with entries denoted by $A=[a_{ij}]$. When $i=j+1$, $a_{ij}=1$, and when $i=j-1$, $a_{ij}=-1$, with all other entries being zero. Determine the eigenvalues of $A$.
To analyze the structure of $A$, consider the following representation for $n \times n$:
$$A = \begin{bmatrix} 0 & -1 & 0 & 0 & \dots & 0 & 0 \\ 1 & 0 & -1 & 0 & \dots & 0 & 0 \\ 0 & 1 & 0 & -1 & \dots & 0 & 0 \\ 0 & 0 & 1 & 0 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \dots & 0 & -1 \\ 0 & 0 & 0 & 0 & \dots & 1 & 0 \end{bmatrix}$$
Eigenvalues were computed for small values of $n$ with the following outcomes:
- For $n=1$, the eigenvalues are $\{0\}$.
- For $n=2$, the eigenvalues are $\{i, -i\}$.
- For $n=3$, the eigenvalues are $\{i\sqrt{2}, -i\sqrt{2}, 0\}$.
However, finding a general expression for the eigenvalues of $A$ remains elusive. Assistance in solving this general case is highly appreciated.
