Vtyjkyuk

Let $A$ be an invertible square matrix in $mathbbR^ntimes n$. 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 $1leq ileq n$ $e_i$ 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 $0leq ileq k$ and $v$ where all coefficients are nonnegative with the possible exception of the coefficient of $v$?

Let
$$A=beginbmatrix 2 & 0 & 0 \ 0 & 1 & 1 \ 0 & 0 & 1 endbmatrix Rightarrow A^k=beginbmatrix 2^k & 0 & 0 \ 0 & 1 & k \ 0 & 0 & 1 endbmatrix$$
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
$$B=beginbmatrix 2 & 0 & 0 \ 0 & 1 & -1 \ 0 & 0 & 1 endbmatrix Rightarrow A^k=beginbmatrix 2^k & 0 & 0 \ 0 & 1 & -k \ 0 & 0 & 1 endbmatrix$$

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 $i<l$.