I am working on a square matrix that is asymmetric and reducible (not strong connected), say $A$, which is an adjacency matrix (all the elements are either $0$ or $1$).

I am wondering if there is a similar conclusion like Perron-Frobenius theorem that the largest eigenvalue of $A$ and the corresponding eigenvector would be non-negative (not all-zero vector).

Is it possible? If it is, how can I proof it? If not, what else requirements should be proposed?

Thank you.

  • $\begingroup$ @saulspatz, I edited my question, $A$ is an adjacency matrix. $\endgroup$
    – Richard
    Apr 17 '18 at 15:54
  • 1
    $\begingroup$ The section titled "Applications" of the Wikipedia article you cited seems to address this question, but it isn't very clear to me. Maybe it will give you someplace to start. $\endgroup$
    – saulspatz
    Apr 17 '18 at 16:12

Take a look at this subsection of the Wikipedia page on the Perron-Frobenius Theorem.


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