Lower bound for the Perron eigen-value of a non-negative, irreducible, integer matrix

[Rob10]

[FONT=MathJax_Math]A is a non-negative, integer, irreducible, [FONT=MathJax_Math]m[/FONT] by [FONT=MathJax_Math]m[/FONT] matrix. It is well known (Perron-Frobenius) that [FONT=MathJax_Math]A[/FONT] has a positive eigen value (denote it by [FONT=MathJax_Math]λ[/FONT]) with a positive eigen vector ([FONT=MathJax_Math]x[/FONT]). It is needed to prove that:[/FONT]

[FONT=MathJax_Math]x_{[FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]a[/FONT][FONT=MathJax_Math]x[/FONT]}[FONT=MathJax_Main]/[/FONT][FONT=MathJax_Math]x[/FONT]_{[FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]n[/FONT]}[FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math]λ[/FONT]^{[FONT=MathJax_Math]m[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT]}[/FONT]​

[FONT=MathJax_Math]x_{[FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]a[/FONT][FONT=MathJax_Math]x[/FONT]} denotes the largest element of [FONT=MathJax_Math]x[/FONT], [FONT=MathJax_Math]x[/FONT]_{[FONT=MathJax_Math]m[/FONT][FONT=MathJax_Math]i[/FONT][FONT=MathJax_Math]n[/FONT]} denotes the smallest element of [FONT=MathJax_Math]x[/FONT].[/FONT]
I proved that when considering A as an adjacency matrix, where there is an edge from state [FONT=MathJax_Math]u to state [FONT=MathJax_Math]v[/FONT], the following holds:[/FONT]

[FONT=MathJax_Math]x_{[FONT=MathJax_Math]v[/FONT]}[FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math]λ[/FONT][FONT=MathJax_Math]x[/FONT]_{[FONT=MathJax_Math]u[/FONT]}[/FONT]​

(it was the hint in this question). I don't see how to proceed from here.

 

[stapel]

Found elsewhere online:

Well, I found the answer. It is easy to prove that $\displaystyle x_v\,\leq \,\lambda x_u$ simply by writing down $\displaystyle Ax\,=\, \lambda x.$ Now, since $\displaystyle A$ is irreducible, there must exist a power $\displaystyle k\, \leq\, m$ such that state $\displaystyle u$ is connected to state $\displaystyle v$ by a path of length $\displaystyle k.$ $\displaystyle A^k$ represents a graph with paths of length $\displaystyle k$ between the states of the $\displaystyle A$-graph. This matrix has eigenvalue $\displaystyle \lambda^k$ with eigenvector $\displaystyle x.$ Thus, we must be able to find indices $\displaystyle u$ and $\displaystyle v$ such that $\displaystyle u$ corresponds to $\displaystyle x_{min}$ and $\displaystyle v$ corresponds to $\displaystyle x_{max}$ for some $\displaystyle k$, so:

$\displaystyle \frac{x_{max}}{x_{min}}\, \leq\, \lambda^k\, \leq\, \lambda^{m-1}$