Consider two $n\times n$ matrices, $A$ and $B(t)$. Here $A$ is time independent, and $B(t)$ is time dependent such that $A$ and $B(t)$ don't necessarily commute. I want to analyze $\exp(A+B(t))$. Ideally, I am looking to expand $\exp(A+B(t))$ in the form,
$$ \exp(A+B(t)) = \exp(A) + \cdots, $$
since I am interested in the difference $\exp(A+B(t))-\exp(A)$. I am also interested in some special forms of $B(t)$:
- $B_1(t) = \sum_{j=1}^p C_l t^l $, where $C_l$ are time independent matrices,
- $B_2(t) = \frac{d}{dt}, $
- $B(t) = B_1(t) + B_2(t)$.
Does anyone have any suggestions for formulas/approaches can be useful here?
Assuming that $B(0) = 0$ (it is the case for $B_1$ and I don't understand what $B_2(t) = \frac{d}{dt}$ means), you can do a Taylor development around $0$ with, $$ \exp(A + B(t)) = \exp(A) + d\exp(A)B(t) + \mathrm{O}(B(t)^2). $$ Notice that for $B = B_1$, then $d\exp(A)B(t) + \mathrm{O}(B(t)^2) = td\exp(A)C_1 + \mathrm{O}(t^2)$.
And for all $H$, $$ d\exp(A)H = \int_0^1 \exp((1 - s)A)H\exp(sA) \, ds, $$ which equals $\exp(A)H$ if $A$ and $H$ commute. I hope it helps you.