I'm trying to understand what should be a simple point in this question: Inverse function theorem for matrices.
Let $A$ be some matrix, and $L(H)$ be some linear function of $H$ (a matrix) with coefficients being powers of $A$. Since we do not have commutativity, $L(H)$ might equal, say, $AH+HA$.
The question is, when is this mapping invertible? According to the answer to the question above, it should only happen when the only solution to $AH+HA=0$ is $H=0$.
Why is this equivalent to being invertible?