Let $G$ be a square matrix of the form $$G=\left[ \begin{array}{cc} A & B \\ C & 0 \\ \end{array} \right]$$ with $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$ and $C \in \mathbb{R}^{m \times n}$, such that all entries of $B$ and $C$ as well as all non-diagonal entries of $A$ are non-negative.

Is it true that $G$ always has an eigenvalue with non-negative real part?

Metzler matrixandHurwitz matrix. – David Handelman Jul 9 at 13:02