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I’m working on contractive systems that have a system of ODE equations. I have an exponential matrix multiply by time that is for a given matrix $A$ I’m getting $e^{At}$. I want to know what are the conditions on $A$ such that the exponential will decay as $t$ goes into infinity?

Thanks in advance.

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You can easily find the answer in more or less any book on ODEs and dynamical systems, such as the one by Teschl.

I think that the decay in matrix norm of $e^{At}$ is equivalent to the fact that all eigenvalues of $A$ have strictly negative real part. But please double check in the book above.

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