Positive eigenvalues and Schur complements For a symmetric matrix,
$$M = \left(\begin{array}{cc}
  A & C\\
  C^{\top} & D
\end{array}\right)$$
it is well known that $M$ is positive definite if and only if $A$ and the Shur complement $M\backslash A = D-CA^{-1}C^T$ are positive definite.
Is there a generalization of this fact, for non-symmetric matrices? Can we claim that:
$$M = \left(\begin{array}{cc}
  A & C\\
  B & D
\end{array}\right)$$
has all eigenvalues with positive real part, if and only if $A$ and $M\backslash A = D - CA^{-1}B$ have eigenvalues with positive real parts too?
I am particularly interested in matrices $M$ which are not symmetric, but for which $B=-C^T$.

There is a counterexample where $A$ and $M\backslash A$ have both positive eigenvalues, but $M$ has an eigenvalue with negative real part, namely
$$
\left(
\begin{matrix}
1 & -2 \\
2 & -2
\end{matrix}
\right)
$$
A has eigenvalue $1$ and $M\backslash A$ has eigenvalue 2, but $M$ has two eigenvalues with real part $-\frac{1}{2}$.
