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Math Help - Inequalities for exponential matrix

  1. #1
    Newbie Sorombo's Avatar
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    Inequalities for exponential matrix

    Hey guys! I've had a couple of troubles trying to show that these propositions are equivalent. Would you mind giving me a hand?

    $\noindent (a) All eigenvalues of A have positive real part \\ \\ (b) For every norm of $M_{n\times n}$. There exist $L>0$ and $\alpha>0$ such such that: $\left \| e^{tA}x \right \|\geq Le^{\alpha}\left \| x \right \| $ \\ For all $t \geq 0 $, $x \in \mathbb{R}^N$ \\ \\(c) There is $a>0$ and a norm $\\ \left \|\  \cdot  \  \right \|_\ast$  such that:  $\left \| e^{tA}x \right \|\geq Le^{\alpha}\left \| x \right \| $ \\ For all $t \geq 0 $, $x \in \mathbb{R}^N$ $

    Regards!
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  2. #2
    MHF Contributor
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    Re: Inequalities for exponential matrix

    Hey Sorombo.

    Are there any results with regard to positive definite matrices that you can use (i.e. relationships to norms)?

    Given that the matrix is positive definite (since it has all positive eigen-values), you should be able to get some results that go with norms.
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