Eigenvalues of a specific matrix I have a block matrix
$$M=\begin{bmatrix}
    I_0&        I_1&        \cdots&     I_1\\
    I_2&        I_0&        \ddots&     \vdots\\
    \vdots&     \ddots&     \ddots&     I_1\\
    I_2&        \cdots&     I_2&        I_0\\
\end{bmatrix}_{n \times n}$$
with
$$I_0=\begin{bmatrix}
    0&      1\\
    1&      0\\
\end{bmatrix}, \qquad
I_1=\begin{bmatrix}
    0&      1\\
    -1&     0\\
\end{bmatrix},\qquad
I_2=\begin{bmatrix}
    0&      -1\\
    1&      0\\
\end{bmatrix}.$$
I want to find all its eigenvalues $\{\lambda_1,\lambda_2,\ldots,\lambda_{2n}\}$, where $\lambda_1 < \lambda_2 < \cdots < \lambda_{2n}$.
Due to the chiral symmetry, we can find $\lambda_i=-\lambda_{2n+1-i}$ for all $i$.
 A: For the signed circulant matrix
$$U:=\left[\begin{matrix}
 & 1 & & &  \\ 
 & &  \ddots & & \\
 & &  & 1 &\\
 -1 &  & & & 
\end{matrix}\right]
\mbox{ in }
M_n(\mathbb{C}),$$
one has
$$M= 1 \otimes I_0 +  (U+U^2+ \cdots + U^{n-1}) \otimes I_1
\mbox{ in }
M_n(\mathbb{C})\otimes M_2(\mathbb{C}).$$
For $\omega:=\exp\frac{i\pi}{n}$,
the unitary matrix $U$ has eigenvalues
$\{ \omega^k : k=1,3,5,\ldots,2n-1\}$
and eigenvectors
$v_k:=[\begin{smallmatrix} 1 & \omega^k & \omega^{2k} & \cdots &\omega^{(n-1)k}\end{smallmatrix}]^{\mathrm{T}}/\sqrt{n}$ in $\ell_2^n$.
Accordingly, the matrix $M$ is decomposed into the direct sum of
$$I_0+ (\omega^k+\omega^{2k}+ \cdots + \omega^{(n-1)k})I_1 
=\left[\begin{matrix}
  0 & \lambda_k\\
  \overline{\lambda_k} & 0
 \end{matrix}\right]
\mbox{ acting on }
\mathbb{C}v_k \otimes \ell_2^2,$$
where
$$\lambda_k=\frac{2}{1-\omega^k}.$$
The eigenvalues of $M$ are
$\pm|\lambda_k|$,  $k=1,3,\ldots,2(n-1)$
with eigenvectors
$ v_k \otimes 
[\begin{smallmatrix} 1 & \pm \mathrm{sgn}(\overline{\lambda_k}) \end{smallmatrix}]^{\mathrm{T}}/\sqrt{2}$ in $\ell_2^n \otimes \ell_2^n$.
Note that since $|\lambda_k|=|\lambda_{2n-k}|$, all eigenvalues
except for $1$ (corresponding to the case when $n$ is odd and $\lambda_n=-1$) have multiplicity $2$.
