Matrix exponential - Knowlpedia

For a square matrix $A$ , its matrix exponential is

e^A=\sum_{n=0}^\infty\frac{A^n}{n!}.

The operator norm bounds this series by $e^{\|A\|}$ , proving absolute convergence. The same definition works for a bounded operator on a Banach space.

Constant-coefficient evolution

Termwise differentiation gives $\frac{d}{dt}e^{tA}=Ae^{tA}=e^{tA}A$ , so $e^{(t-s)A}$ is the propagator of $y'=Ay$ . Also $e^{tA}e^{sA}=e^{(t+s)A}$ , hence $e^{tA}$ is invertible. For distinct matrices, $e^{A+B}=e^Ae^B$ is guaranteed when $AB=BA$ and is false in general without a commutation hypothesis.