Solve matrix exponential power series The matrix exponential is defined as
$$ e^A  = \sum_{n=0}^\infty \frac{1}{n!} A^n  $$
However I would like to solve something similar:
$$ B = \sum_{n=1}^\infty \frac{1}{n!} A^{n-1}  $$
(NOTE: starting index and the power)
I can transform that into
$$ B = A^{-1}\sum_{n=0}^\infty \frac{1}{n!} A^{n} -I = A^{-1} \left(e^A - I \right) $$



*

*That works. But when $A^{-1}$ does not exist ($A$ is not invertible) it does not work. 

*Is it possible to make it work for general $A$? Has anyone thoughts on that? Thanks.

Your matrix B is in literature called phi-function $\varphi_1$. The power series expansion of the general form is
\begin{align}
   \varphi_k (A) := \sum_{n=0}^\infty \frac{A^n}{(n+k)!}
\end{align}
If an approximate solution is sufficient a Pade approximation $\varphi_1(A) \approx (1-A/2)^{-1}$ would be a possible solution.
