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Consider a positive linear dynamical system. $\frac{dx}{dt}=Ax$, where $A$ is a quasipositive/Metzler/essentially nonnegative matrix. By its properties, the vector $x$ will remain positive for all time $t$ if it starts from a positive vector $x(0)$. Let the unique largest eigenvalue (which is always purely real) of $A$ be negative. Then the vector $x$ will converge surely to 0. Usual linear dynamical systems theory tells us that $||x(t)||\le ||x(0)|| e^{\lambda_{max} t}$.

So far so good. But there is another property to Metzler matrices: the eigenvector corresponding to the maximum eigenvalue is positive. I am wondering if there is any special interpretation to this normalised eigenvector. Positive eigenvectors in the context of PageRank algorithm stand for relative importance of different pages. What does it stand for in the context of postive linear systems?

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  • $\begingroup$ Don't you need some irreducibility assumptuion? $\endgroup$ – András Bátkai Feb 13 '14 at 19:40
  • $\begingroup$ You can check some details about this in our lecture notes: isem17.unisa.it/w/images/4/4c/Isem1314_PartI.pdf $\endgroup$ – András Bátkai Feb 13 '14 at 19:51
  • $\begingroup$ @AndrásBátkai: Yes, the matrix is irreducible as well. I can't see if there is any probabilistic interpretation to the positive eigenvector. Will be glad to hear from you. $\endgroup$ – Sultan Feb 15 '14 at 11:42
  • $\begingroup$ isn't it the same, as pagerank: the entries tell which state of the system we are more likely to be in (think of moving from state-to-state as time passes, then which state are we most likely to be in will be given by the entries of the Perron eigenvector) $\endgroup$ – Suvrit May 15 '14 at 13:47
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Well, one obvious observation is that the matrix $A-\lambda_{max}$ has largest eigenvalue 0. Hence, if the matrix is irreducible (as András says), then the semigroup generated by $A-\lambda_{max}$ converges to the orthogonal projector onto the first eigenspace -- i.e., onto the space spanned by your Perron eigenvector of $A$.

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  • $\begingroup$ Thanks, this is true but this does not make use of the fact that it is a positive vector. I was wondering if there is any probabilistic interpretation to the eigenvector. $\endgroup$ – Sultan Feb 14 '14 at 13:15
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I suggest you to check these lecture notes.

Proposition 5.3.2 is the statement and possible interpretations you find in Section 6.3.

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