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Let $A$ be a stochastic matrix, $q\in (0,1)$. How to bound $n$ such that $$q^n A^n e^A \leq e^A$$

Note that here $e^A$ is the matrix exponential, and $\leq$ is taken entrywise.

To be clear, what I want is some $N$, in terms of (for instance) entries of $A$ and $q$, such that for any $n\geq N$, $q^n A^n e^A \leq e^A$.

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  • $\begingroup$ If $A$ is primitive, then you can get an estimate of how close powers of $A$, and also the tail of the exponential is to the limiting idempotent, and this would give something. But you can't expect too much, since there is a constant in the rate of convergence ... $\endgroup$
    – David Handelman
    Commented Jul 12, 2016 at 23:01

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