limit of an exponentiated sum

Question

equation 2 of this paper states that

$$ \lim_{t \rightarrow \infty} \sum_{i=0}^{t-1}(\alpha S)^i = (I - \alpha S)^{-1} $$

In this case $0 < \alpha < 1$ and the entries of $S$ are in the range $[-1, 1]$. How can I understand why this is true?

Answer 1

You wrote: "the entries of $S$ are in the range $[−1,1]$."

But in the paper we have the eigenvalues of $S$ are in $[−1,1]$.

In this case each eigenvalue of $ \alpha S$ are in the intervall $(-1,1)$, since $0< \alpha <1.$

Hence the spectral radius of $ \alpha S$ is $<1,$ therefore the geometric series $\sum_{i=0}^{\infty}(\alpha S)^i$ is convergent and $ = (I - \alpha S)^{-1}.$