## Definition

Let $G=(V,E,A)$ be a **strongly connected** directed graph, where $V=\{1,2,...,n\}$ denotes the vertex set, $E$ is the edge set, and $A$ is the associated adjacency matrix with $0-1$ weighting, that is $a_{i,j}=1$ if $(j,i)\in E$, and $a_{i,j}=0$ otherwise.

$B$ and $D$ are two diagonal matrices, where $b_{ii}=\sum_{j=1}^na_{i,j}$ and $d_{ii}=\sum_{j=1}^na_{j,i}$. In other words, the diagonal entries of $B$ are the row sum of $A$, and the diagonal entries of $D$ are the column sum of $A$.

## Problem

Now define a new matrix
$$M =
\begin{bmatrix}
B-A, & -A \\\\
A-B, & D
\end{bmatrix}\in \mathbb{R}^{2n\times 2n}$$ Since the column sum of $M$ are identical zeros, zero must be one of its eigenvalue. Can I **claim** that the rest eigenvalues all have positive real parts?

I tried many numerical examples, the rest eigenvalues all have positive real parts. Anyone can help prove or disprove the above claim? (Gershgorin Circle Theorem does not apply here because $M$ is not diagonally dominate)

Some facts: Both $(B-A)$ and $(D-A)$ have exactly one zero eigenvalue and all the rest eigenvalues lie in the open right half complex plane because the directed graph is strongly connected. In particular, $(B-A)$ is called the Laplacian matrix of the graph.