# The problem about the eigenvalues of Metzler matrix in a special form

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?

• how can this be an upper triangular matrix if $C$ is nonzero? – Carlo Beenakker Jul 9 at 8:51
• It has be revised because of the Inaccurate description. – Popcorn Jul 9 at 9:15
• take $A=B=C=0$, then there are no positive eigenvalues, and yet $A$ is a Metzler matrix. – Carlo Beenakker Jul 9 at 10:02
• It would help if you defined Metzler matrix and Hurwitz matrix. – David Handelman Jul 9 at 13:02