Let be $A$ a $n\times n$ fully indecomposable matrix. Let be $$\Sigma=\{x\in\mathbb{R}^n: x_i>0, \ i=1,\dots,n, \ \|x\|_1=1\}.$$ I would like to show that it exist $\varepsilon>0$ such that if $y=Ax$ with $x\in\Sigma$ then $y_i>\epsilon \ \ \forall i=1,\dots,n.$
A non negative matrix $A\in\mathbb{R}^n$ is said fully indecomposable if it doesn't exist permutation matrices $P,Q$ such that $$PAQ=\begin{bmatrix}X &0\\Y & Z\end{bmatrix}$$ where $X,Z$ are square matrices.
Any hint?