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Assume that $A$ is the adjacency matrix of a strongly connected directed graph, that is, $A$ is non-negative and irreducible. Let $$L_\text{in}=D_\text{in}-A',\;L=D_\text{in}-A'+D_\text{out}-A$$ where $D_\text{in}$ and $D_\text{out}$ are the in and out degree matrix, respectively. i.e. $$D_\text{in}=\text{diag}(A'\mathbb{1}),\; D_\text{out}=\text{diag}(A\mathbb{1})$$ where $\mathbb{1}$ is a vector with all elements be 1 and diag() is the Matlab function diag(). I wonder if the matrix $$M = L'L_\text{in}+L_\text{in}'L$$ is positive semi-definite?

I have tried several $A$ using Matlab and the results are all positive semi-definite, so I think it is true. Actually, $L$ is positive semi-definite and each eigenvalue of $L_\text{in}$ has a positive real part. Besides, both $L$ and $L_\text{in}$ have zero row sums. Is there any suggestion for a proof or a counterexample? Thank you very much!

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  • $\begingroup$ 1) Prime denotes transpose? 2) Are your matrices real valued? $\endgroup$
    – lcv
    Jul 13, 2017 at 9:15
  • $\begingroup$ @lcv 1) Yes. 2) Yes. $\endgroup$
    – Jiaqi
    Jul 13, 2017 at 14:27

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