Derive an expression for the inverse of $\textrm{exp}(A)=I+A+A^2/2!+A^3/3!+\dots$

Question

Let $\textrm{exp}:M_{2\times2}(\mathbb{R})\rightarrow M_{2\times2}(\mathbb{R})$ denote the function on the space of $2\times2$ matrices defined by

$\textrm{exp}(A)=I+A+A^2/2!+A^3/3!+\dots$.

Derive an expression for the inverse of exp.

I gather that the inverse ought to resemble some form of $\ln{x}$, so the solution ought to look like some Taylor expansion of this function in terms of matrices. Proving this rigorously is giving me trouble though.

Any help appreciated!

Answer 1

It is true that $e^{A+B}=e^{A}e^{B}$, if $AB=BA$. Hence, for $B=-A$ it is clear that the inverse of $e^{A}$ is $e^{-A}$. I hope now that you can easily calculate what you are asking, through substitution.

Answer 2

Let a matrix $B = f(A) = \exp A$, then the inverse function can be understood as a function of a matrix $B$, namely $g(B) = f^{-1}(B) = A$. The required function $g(B)$ can be understood as the "matrix logarithm" as generalization of the scalar logarithm. Details on when the matrix logarithm exists and how its series representation looks like can e.g. be found here (Wikipedia).