Can any one prove the following inequality $$||e^{Pt}||\leq e^{t\alpha{(P)}}\sum_{k=0}^{r-1}\frac{(||P||\sqrt{r}\,t)^k}{k!}$$, where $r$ is the order of the matrix $P$ and $\alpha(P)$ be the maximum eigenvalue of the matrix $P.$ With details.

  • $\begingroup$ Try the Jordan canonical form. You should add the linear algebra tag. $\endgroup$
    – Michael
    Dec 3, 2013 at 12:52


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