I have a block matrix $$ M=\left[ \begin{matrix} I_0& I_1& \cdots& I_1\\ I_2& I_0& \ddots& \vdots\\ \vdots& \ddots& \ddots& I_1\\ I_2& \cdots& I_2& I_0\\ \end{matrix} \right]_{n\times n} $$ with $$ I_0=\left[ \begin{matrix} 0& 1\\ 1& 0\\ \end{matrix} \right], I_1=\left[ \begin{matrix} 0& 1\\ -1& 0\\ \end{matrix} \right], I_2=\left[ \begin{matrix} 0& -1\\ 1& 0\\ \end{matrix} \right]. $$
$$M=\begin{bmatrix} I_0& I_1& \cdots& I_1\\ I_2& I_0& \ddots& \vdots\\ \vdots& \ddots& \ddots& I_1\\ I_2& \cdots& I_2& I_0\\ \end{bmatrix}_{n \times n}$$
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 solvefind all its eigenvalues $\{\lambda_1,\lambda_2,\ldots,\lambda_{2n}\}$, where $\lambda_1<\lambda_2<\cdots<\lambda_{2n}$$\lambda_1 < \lambda_2 < \cdots < \lambda_{2n}$.
Due to the chiral symmetry, we can find $\lambda_i=-\lambda_{2n+1-i}, \forall i$$\lambda_i=-\lambda_{2n+1-i}$ for all $i$.
