[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]
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]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.