# Transition Matrix Operation in Stoikov's Micro Price Paper

Sasha Stoikov's paper provides an interesting finite state approach to modeling the mid. It makes good sense to me except one property.

On page 7 of the linked paper, $$G^1(x)=\left(\sum_s\mathbf{Q}^{s-1}\mathbf{R}\right)\mathbf{K}=(1-\mathbf{Q})^{-1}\mathbf{R}\mathbf{K},$$ where $$\mathbf{Q}$$ is the $$nm\times nm$$ transition matrix of the transient states and $$\mathbf{R}$$ the $$nm\times 4$$ transition matrix of the absorbing states. In my mind, the full transition matrix should look like the following:

$$\mathbf{T}=\left(\begin{array}{cc}\mathbf{Q}&\mathbf{R}\\\mathbf{0}&\mathbf{I}\\\end{array}\right),$$

which does agree with Sasha's own beamer.

Question: What is the property applied here so that $$\sum_s\mathbf{Q}^{s-1}\mathbf{R}=(1-\mathbf{Q})^{-1}\mathbf{R}$$?