Let $A_1,A_2 \in \mathbb{R}^{n \times n}$. Construct the block matrix $A$ as follows: $$A: = \left[ {\begin{array}{*{20}{c}} 0&{{A_1}}\\ {{A_2}}&0 \end{array}} \right]$$ My observation is that matrix $A$ cannot have all its eigenvalues in the open left-half plane. In other worlds, if all eigenvalues of $A$ have non-positive real parts, then all of them are imaginary.
I greatly appreciate it if someone can give me some idea/intuition why this happens. Thanks
The characteristic polynomial of this matrix is of the form $p(\lambda) = q(\lambda^2)$, where $q$ is the characteristic polynomial of $A_1A_2$. Therefore all roots come in pairs.