Eigenvalues for a block matrix with Toeplitz tridiagonal sub-matrix

Question

Given a matrix $M \in \mathbb{R}^{(2N, 2N)}$ for some $N \in \mathbb{Z}, N > 2$

$$M = \begin{pmatrix}\textbf{0}&I\\A&\textbf{0}\end{pmatrix},$$

where $\textbf{0} \in \mathbb{R}^{(N, N)}$ is a zeroed matrix, $I \in \mathbb{R}^{(N, N)}$ is the identity matrix and $A \in \mathbb{R}^{(N, N)}$ is a negative-definite Toeplitz Tridiagonal matrix with entries: $-2$ on the diagonal and $1$ on the subdiagonal and superdiagonal, i.e.

$$ A = \begin{pmatrix} &-2 &1 &0 &\ldots &\ldots &\ldots &0 \\ &1 &-2 &1 &\ddots &\ddots &\ddots &\vdots \\ &0 &1 &-2 &1 &\ddots &\ddots &\vdots \\ &\vdots &\ddots &\ddots &\ddots &\ddots &\ddots &\vdots \\ &\vdots &\ddots &\ddots &1 &-2 &1 &0 \\ &\vdots &\ddots &\ddots &\ddots &1 &-2 &1 \\ &0 & \ldots &\ldots &\ldots &0 &1 &-2 \end{pmatrix}$$

Show that all the eigenvalues of $M$ are imaginary.

Answer 1

Hint: Eigenvalues are $\pm \sqrt{-2+2\cos(\theta_j)}, \quad \theta_j=j\pi/(N+1), \quad j=1,\ldots,N$.