Let $A$ be an invertible square matrix in $\mathbb{R}^{n\times n}$. Let $\lambda$ be the unique eigenvalue of $A$ with the largest norm. Assume that we have two good properties, namely $\lambda$ is real and positive and its eigenspace has dimension 1. Let $v$ be one of the unit eigenvectors.
We would like to examine whether both of the two properties are correct:
a) For any $1\leq i\leq n$ $e_i$ can be written as a linear combination of column vectors of column vectors of $A^k$ for positive $k$ and $v$ where all coefficients are nonnegative with the possible exception of the coefficient of $v$?
b) Does there exist a $k$ such that for all $l>k$ any column vector of $A^l$ can be written as a linear combination of column vectors of $A^i$ where $0\leq i\leq k$ and $v$ where all coefficients are nonnegative with the possible exception of the coefficient of $v$?
Let $$A=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix} \Rightarrow A^k=\begin{bmatrix} 2^k & 0 & 0 \\ 0 & 1 & k \\ 0 & 0 & 1 \end{bmatrix}$$ Then the largest eigenvalue of $A$ is 2 and its corresponding eigenspace has dimension 1.
a. One takes $\nu=e_1$ and notes that $e_3$ cannot be written as a combination of the second and third columns of $A^k$ with nonnegative coefficients.
b. Let $$B=\begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{bmatrix} \Rightarrow A^k=\begin{bmatrix} 2^k & 0 & 0 \\ 0 & 1 & -k \\ 0 & 0 & 1 \end{bmatrix}$$
Again $B$ satisfies the required conditions. However, the third column of $A^l$ cannot be written as a combination of the second and third columns of $A^i$ for $i<l$.