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 $i$.



Eigenvalues of a specific matrix
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 $2$.


