Let P be a irreducible stochastic matrix and Q be a matrix obtained by making one positive entry of P to be zero and remaining terms are same as P. If 1,t are the largest and second largest positive eigenvalues of P and s is the largest eigenvalue (spectral radius) of Q, is it true that t ≤ s < 1?
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1Your question doesn’t make sense because $t$ can be non-real. – user1551 Oct 26 '23 at 10:29
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We assume that P has a positive real eigenvalue other than 1. – Haritha Cheriyath Oct 27 '23 at 11:11
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So, if the eigenvalues of $P$ are $1,,\pm0.9i$ and $0.8$, do you mean the ”second largest” eigenvalue of $P$ is $0.8$? – user1551 Oct 27 '23 at 11:14
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Yes,it is the second largest among the positive eigenvalues, assuming there is at least one less than 1. – Haritha Cheriyath Oct 28 '23 at 04:23
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It is true that $\rho(Q)<1$, for the reason described here. – user1551 Oct 28 '23 at 09:18
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Yeah, do we know anything about the other inequality? – Haritha Cheriyath Oct 28 '23 at 09:28