Laurent series of matrix function

Question

I need to find the Laurent series of a matrix function around $z=\infty$, but I lack the math knowledge. I can't find a way to get it with WolframAlpha or sympy. The equation is simply

$$(z - \mathbf{V}(z))^{-1} $$

For a scalar $v$ WolframAlpha gives me the series

$$\frac{1}{z-v} = \frac{1}{z} + \frac{v}{z^2} + \mathcal{O}(z^{-3}).$$

I assume this can be obtained from the Geometric series? How do I get the generalization for matrices?

Answer 1

@user550103 gave the right hint in the comment, the Neumann series

$$ {(1 - \mathbf{T})}^{-1} = \sum_{k=0}^{\infty}\mathbf{T}^k$$

can be applied to abtain the expansion:

$$ {(z - \mathbf{V}(z))}^{-1} = z^{-1}{(1 - \mathbf{V}(z)/z)}^{-1} = z^{-1}\sum_{k=0}^{\infty}\frac{\mathbf{V}^k(z)}{z^k} = \sum_{k=0}^{\infty}\frac{\mathbf{V}^k(z)}{z^{k+1}}. $$