Let's say a matrix M is composed of:

\begin{bmatrix} A & B \\ C & D \end{bmatrix} where $A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{m \times n},$ and $D \in \mathbb{R}^{m \times m}$.

I would like to determine whether all eigenvalues in this matrix are positive (or there is an imaginary part).

I know that $A$ and $D$ are invertible. While I tried to make use of Schur complement, it requires $B = C^T$, which does not satisfy in my case. Do you have any recommendation on how to proceed from this step? Thank you very much!


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