Magnus expansion

Remarks

The series converges, and gives the solution, when $\int_0^t \norm{A(s)}_2\, ds < \pi$, where $\norm{\cdot}_2$ is the spectral norm; this is a sufficient condition. Example: if $A(t_1)$ and $A(t_2)$ commute for all $t_1, t_2$, all terms with $k \ge 2$ vanish and $Y(t) = \exp\left(\int_0^t A\right)$; for $A(t) = a(t) N$ with a fixed matrix $N$ and scalar $a$, this gives $Y = e^{(\int_0^t a) N}$. Without commutation, $\exp\left(\int_0^t A\right)$ is in general not the solution, and $\Omega_2$ is the first correction. The reason is that the derivative of $\exp(\Omega_1(t))$ need not be $A(t) \exp(\Omega_1(t))$ when $A(t)$ does not commute with $\Omega_1(t)$. Let $A(t)$ lie in a matrix Lie algebra for all $t$, such as the skew-symmetric matrices. Then every truncation of the series also lies in it, so its exponential lies in the corresponding group.