# Inequalities for exponential matrix

• Jun 4th 2013, 08:40 PM
Sorombo
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?

$\displaystyle$\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!
• Jun 5th 2013, 06:16 PM
chiro
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.