Consider a full rank matrix $A \in \mathbb{R}^{n \times n}$ and its determinant $det(A) > 0$. For a vector $x \in \mathbb{R}^n$ with positive elements, we are interested in when the sequence $\{x, x + Ax, x + Ax + A^2x, \cdots\}$ will convergence to a vector
$c \begin{bmatrix} \frac{1}{n} \\ \frac{1}{n} \\ \vdots \\ \frac{1}{n} \\ \end{bmatrix}\space \text{where} \space c > 0$.
Should we discuss the largest singular value $\sigma_1$, if $\sigma_1 > 1$, then $A^n$ divergence?
Actually, I have no idea but maybe we need to study the property of $I + A + A^2 + A^3 + \cdots + A^m + \cdots$.
Is there any reference materials about it?
Let $A$ be a square matrix of order $n$. The norm of $A$, denoted by $||A||$ is defined as the normed operator defined by $A$ as an operator. Let $||A||\lt1$, then $I+A+A^2+...$ convergent to $(I-A)^{-1}$. If $\sigma_{1}{\gt 1}$ then $||A||\gt1$ and $A^{n}$ divergent. Note that we can represent $A=PTP^{-1}$, where $T$ is an upper triangular matrix having $\sigma_{1}$ in the diameter. So the limit $\lim_{n\to\infty}A^{n}=P(lim_{n\to\infty}T^{n})P^{-1}$ do not exist. Let $A=Diam(\frac{1}{2} , \frac{1}{2} , ...,\frac{1}{2})$ . Then $B=I+A+A^2+...=Diam(2 , 2 ,...,2)$ and $B[\frac{1}{2n} , \frac{1}{2n} ,...,\frac{1}{2n}]^t=[\frac{1}{n} , \frac{1}{n} ,...,\frac{1}{n}]^t$. Note that we consider $x=[\frac{1}{2n} , \frac{1}{2n} ,...,\frac{1}{2n}]^t$ and $c=1$.