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Let $X,Y\in\mathbb{R}^{n\times n}$ and $a,\theta>0$. Define \begin{equation} A\triangleq\left[\begin{array}{cc}2(\cos\theta) I_n-a X&a Y-I_n\\I_n&0_n\end{array}\right]\in\mathbb{R}^{2n\times 2n}. \end{equation} Let $P(\lambda)$ be the charactristic polynomial of $A$, which is given by \begin{equation} P(\lambda)\triangleq\det\left((\lambda^2-2\lambda\cos\theta +1)I_n+a(\lambda X-Y)\right). \end{equation} I am trying to find conditions on $X,Y$, $a$, and $\theta$ such that all eigenvalues of $A$ are inside open unit disk.

For the case $n=1$, $P(\lambda)=0$ is a quadratic equation in $\lambda$ and I can find the conditions. But I have no idea how to find conditions for cases where $n>1$. Any idea? Thanks

Yasi
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    Your matrix is called ''upper anti-triangular'' block matrix, the general form of which is: $\begin{bmatrix} A & B \ C & 0 \end{bmatrix}$. The following question may be useful: https://math.stackexchange.com/questions/345681/eigenvalue-of-anti-triangular-block-matrix-skew-matrix. – PseudoRandom Jun 30 '17 at 10:07

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