Power series for a matrix inverse

Question

Is there a power series expansion for a matrix inverse of the form $$\left(\frac{1}{m}I+A\right)^{-1} \mbox{ where $m$ is a scalar?}$$

$A$ is not invertible but the expression above is defined. I don't want to embed $m$ into the $A$ matrix as I want the result to have $m$ in it explicitly.

Thanks!

Answer 1

We have $$ \left(\frac{1}{m}I+A\right)^{-1}=m(I+mA)^{-1}=m\sum_{n=0}^\infty (-1)^nm^nA^n=\sum_{n=0}^\infty (-1)^nm^{n+1}A^n. $$ The series above (known as Neumann series) converges whenever $$ \lvert m\rvert\cdot\|A\| <1. $$

Answer 2

As far as I remember, these type of sums are called Neumann's series. $\frac{1}{I-M}=\sum_0^\infty M^k$ under the "right" assumption on $M$. Operator $M$ should be bounded, you have that for free since you deal with matrices, and series converges in norm.