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$N\times N$ matrix $L$ can be partitioned into $p\times p$ blocks,

$L=\left[\begin{array}{cccc} L_{11} & L_{12} & \cdots & L_{1p}\\ L_{21} & L_{22} & \cdots & L_{2p}\\ \vdots & \vdots & \ddots & \vdots\\ L_{p1} & L_{p2} & \cdots & L_{pp} \end{array}\right] $.

Here each row sum of $L_{ij}$ is zero. The square block matrix $L_{ii}$ has some special properties: $l_{mn}\leq 0$, when $m\neq n$, and $l_{mm}=-\sum_{n}{l_{mn}}$。Denote $D=diag\{D_1,D_2,\ldots,D_p\}=diag\{d_1,d_2,\ldots,d_N\}$, where $d_i\geq0$. \begin{equation*} \hat{L}=L+D=\left[\begin{array}{cccc} \hat{L}_{11} & L_{12} & \cdots & L_{1p}\\ L_{21} & \hat{L}_{22} & \cdots & L_{2p}\\ \vdots & \vdots & \ddots & \vdots\\ L_{p1} & L_{p2} & \cdots & \hat{L}_{pp} \end{array}\right], \end{equation*}
There is only one non-zero $d_i>0$ in $D_k\,(k =1,2,\ldots,p)$ such that the eigenvalues of $\hat{L}_{ii}$ are all with positive real parts.
The question is under what condition the eigenvalues of $\hat{L}$ are all with positive real parts?

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