While reading a paper about "Finding Exponential Product Formulas of Higher Order", I come across the statement that by the well-known trotter decomposition
$1$) $e^{x(A+B)} = e^{xA}e^{xB}+O(x^2)$,
where later the author states that mathematicians put 1) in the form:
$2$) $e^{xA}e^{xB}=e^{x(A+B)+O(x^2)}$.
I am trying to understand why equation $2$ is true. Expanding on both sides gives
$LHS = (I + xA + x^2/2A^2 + \cdots)(I + xB + x^2/2B^2 + \cdots)$=
$= I + x(A+B)+x^2/2(A^2+B^2+2AB) + \cdots$
whereas on the right I'm not sure what it means to take the exponential of a polynomial added to a matrix $x(A+B)$. I also do not understand why the remaining terms in $LHS$ for power larger than or equal to 3 are absorbed in the big-oh notation.
Moreover I'm not sure what is going on notation-wise with equation 1), as the left hand side is a matrix, while the right is a matrix added to a set of polynomials?
Edit: Additionally, it seems like in the Big O notation, that we are treating matrices as constants (sort of)? I think this might be the heart of my confusion.
Best Answer
On the right is $$\frac{x^2}2(A+B)^2=\frac{x^2}2(A^2+AB+BA+B^2)$$ which is different because in general $AB\neq BA$. So you need some $O(x^2)$ correction term.