Positive matrix and positive vector Let $A \in \mathbb{R}^{n \times n}$ be a non-negative matrix, i.e. $A_{i,j} \geq 0$ $\forall (i,j)$.
Let $x \in \mathbb{R}^n \setminus \{0\}$ be a non-negative vector, i.e. $x_i \geq 0$ $\forall i$.
Assume $A x > x$ (componentwise), that is $(Ax)_i > x_i$ $\forall i$.
Prove that the following componentwise inequality holds: $$A \left( A x - x\right) > Ax - x.$$
Comment. This is what I tried out. The assumption is equivalent to $(A-I)x > 0$, while the thesis is equivalent to $(A-I)^2 x > 0$. Notice that $A-I$ is Metzler. Now, by contradiction, let $x \neq 0$ such that $(A-I)^2 = 0$, i.e., $x$ belongs to the null space of $(A-I)^2$. Does this imply that $x$ also belongs to the null space of $A - I$?
 A: Here is a counterexample: 
$$A=\begin{pmatrix}0.86 & 0.88 \\ 0.92 & 0.17\end{pmatrix}, \qquad x = \begin{pmatrix} 69 \\11 \end{pmatrix}$$
Indeed, 
$$y:=Ax-x = \begin{pmatrix} 69.02 \\    65.35 \end{pmatrix} -\begin{pmatrix} 69 \\11 \end{pmatrix} = \begin{pmatrix} 0.02 \\    54.35 \end{pmatrix} >0$$
but 
$$Ay  = \begin{pmatrix} 47.8452  \\ 9.2579  \end{pmatrix} \not> y$$

The following code (Scilab)  finds such examples in spades. 
for i=1:100
    A=rand(2,2,"uniform")
    for j=1:100
        x=rand(2,1,"uniform") 
        y=A*x-x
        if min(y)>0 & min(A*y-y)<0 
            disp(A*y-y,x,A)    
        end 
    end
end

A: Here are all the counterexamples with dimension $n = 2$ and entries in $\{0,1,2\}$.
[[0, 1], [0, 2]] [0, 1]
[[0, 1], [0, 2]] [0, 2]
[[0, 1], [1, 1]] [1, 2]
[[0, 1], [1, 2]] [0, 1]
[[0, 1], [1, 2]] [0, 2]
[[0, 1], [2, 2]] [0, 1]
[[0, 1], [2, 2]] [0, 2]
[[0, 2], [0, 2]] [0, 1]
[[0, 2], [0, 2]] [0, 2]
[[0, 2], [1, 1]] [1, 2]
[[0, 2], [1, 2]] [0, 1]
[[0, 2], [1, 2]] [0, 2]
[[0, 2], [2, 2]] [0, 1]
[[0, 2], [2, 2]] [0, 2]
[[1, 1], [1, 0]] [2, 1]
[[1, 1], [2, 0]] [2, 1]
[[2, 0], [1, 0]] [1, 0]
[[2, 0], [1, 0]] [2, 0]
[[2, 0], [2, 0]] [1, 0]
[[2, 0], [2, 0]] [2, 0]
[[2, 1], [1, 0]] [1, 0]
[[2, 1], [1, 0]] [2, 0]
[[2, 1], [2, 0]] [1, 0]
[[2, 1], [2, 0]] [2, 0]
[[2, 2], [1, 0]] [1, 0]
[[2, 2], [1, 0]] [2, 0]
[[2, 2], [2, 0]] [1, 0]
[[2, 2], [2, 0]] [2, 0]
Total found: 28

Python code:
def find_counterexamples(lower, upper) :
    # Matrix A:
    #   a b
    #   c d
    # Vector x: x, y
    count = 0
    for a in range(lower, upper) :
        for b in range(lower, upper) :
            for c in range(lower, upper) :
                for d in range(lower, upper) :
                    for x in range(lower, upper) :
                        for y in range(lower, upper) :
                            if is_valid_counterexample(a,b,c,d,x,y) :
                                print [[a,b],[c,d]], [x,y]
                                count += 1
    print "Total found: "
    print count
    print "\n"

def is_valid_counterexample(a,b,c,d,x,y) :
    x2 = (a - 1) * x + b * y
    y2 = c * x + (d - 1) * y
    x3 = (a - 1) * x2 + b * y2
    y3 = c * x2 + (d - 1) * y2
    if x2 <= 0 or y2 <= 0 :
        return False
    elif x3 > 0 and y3 > 0 :
        return False
    else :
        return True

lower = 0
upper = 1 + int(raw_input("upper: "))
find_counterexamples(lower, upper)

