Let $X\in\mathbb{C}^{n\times n}$ and $a,\theta>0$. Define \begin{equation} A\triangleq\left[\begin{array}{cc}2(\cos\theta) I_n-a {\rm Re~}(e^{\jmath\theta}X)&a {\rm Re~}X-I_n\\I_n&0_n\end{array}\right]\in\mathbb{R}^{2n\times 2n}.\hspace{.8in}(1) \end{equation}
I am trying to find conditions on $X$, $a$, and $\theta$ such that all eigenvalues of $A$ are inside the unit disk.
Below is what I have learned so far:
Assume $A\triangleq \left[\begin{array}{cc}A_1&A_2\\I_n&0_n\end{array}\right]$. Then, the following equations hold: \begin{align} {\rm det~} (\lambda I_{2n}-A)&={\rm det~}\left(\lambda^2I_n-\lambda A_1-A_2\right),\hspace{2in}(2)\\ {\rm det~} (\lambda I_{2n}-A)&={\rm det~}\left(\lambda I_n-A_1\right){\rm det~}\left(\lambda I_n-(\lambda I_n-A_1)^{-1}A_2\right).\hspace{.65in}(3) \end{align} I am not sure if these equations are helpful. Using (2), it follows from (1) that \begin{equation} {\rm spec}(A)=\bigg\{\lambda\in\mathbb{C}:\det\big(\left[\lambda^2-2\lambda\cos\theta+1\right]I_n+a\left[\lambda{\rm Re~}(e^{\jmath\theta}X)-{\rm Re~}X\right]\big)=0\bigg\}, \end{equation} For $n=1$, the eigenvalues can be calculated. However, for $n>1$, I have no idea how to proceed. Any idea or suggestion will be appreciated. Thanks