Can someone give me an idea for the proof that for every $t\in \mathbb{C}$ we have $e^{tA}\cdot A = A \cdot e^{tA} =$ ? I couldn't find a counterexample, so my gues is, that it would be true, but I'm not sure even how to begin the proof.
Does the exponential of a matrix commute with the matrix?
7
$\begingroup$
matrices
analysis
matrix-exponential
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4It suffices to observe that the subspace of all matrices commuting with a fixed matrix is closed. – 2011-11-12
1 Answers
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$e^{tA}\cdot A = \left(\sum_{k=0}^\infty \frac{t^kA^k}{k!}\right)\cdot A$ $= \sum_{k=0}^\infty \frac{t^kA^{k+1}}{k!}$ $= A \cdot \left(\sum_{k=0}^\infty \frac{t^kA^k}{k!}\right)$ $= A \cdot e^{tA}$
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1wow, actually that simple... – 2011-11-12