Given a matrix $A$, the Neumann series https://en.wikipedia.org/wiki/Neumann_series is defined as $$M=\sum_{n=0}^\infty A^n.$$ For a converge $M$, how to calculate the singular value? Is it possible to express the singular values of $M$ by the singular values of $A$?
What I have done: Since $M$ is converge. $M=(1-A)^{-1}$. We know the singular value of an inverse matrix is the reciprocal of the original matrix. But what's the relationship between the singular values of $A$ and $1-A$?
If not easy, can we give some bound of the singular values with $A$'s singular values?