# Eigenvalues of weighted Laplacian

Let $L_{n \times n}$ be a Laplacian matrix of a directed graph, for example, $$L = \begin{bmatrix} 2 & -1 & -1\\ 0 & 1 & -1\\ -1 & 0 &1 \end{bmatrix}.$$ Gersgorin disc theorem tells us that all the eigenvalues of $L$ is on the closed right half plane. If $A_i \in R^{2 \times 2}$ are positive definite matrices, can we prove that the eigenvalues of $\mathrm{diag}([A_1,A_2,\cdots,A_n])(L\otimes I_2)$, for example, $$L = \begin{bmatrix} 2A_1 & -A_1 & -A_1\\ 0 & A_2 & -A_2\\ -A_3 & 0 & A_3 \end{bmatrix}$$ are also on the closed right half plane? Thank you very much.

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