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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?

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The condition you have stated can be rewritten as $\ker(L) = \{0\}$, ie $L$ is injective. If the domain and the range of $L$ have the same (finite) dimension, then this implies that $L$ is also surjective, and thus invertible.

Obviously if you are in an infinite-dimensional space this no longer applies, but since you were speaking of matrices there shouldn't be problems.

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