I have the following problem. Imagine this system of recurrences:
$$\vec P_{t+1}= \begin{bmatrix} -1 && 2 \\ 2&& -1 \end{bmatrix} \vec P_t$$
I'm interested to know if $p_1$, the first element of $P_t$, has a positive feedback. That is, if we increase $p_{1,t}$ this will generate an greater increase some time after in $p_{1,>t}$.
At first glance, it is unlikely that there is a positive feedback because the diagonals are negative. However, I can see that $p_1$ increases $p_2$ in step $t+1$, which will increase $p_1$ in $t+2$. Thus $p_1$ has positive feedback mediated through $p_2$.
How can I write this intuition as a mathematical condition for any matrix?
I understand that for the 2x2 case I can state that the antidiagonal should be all positive or all negative. But when considering 3x3 or 4x4 matrices, there are way more combinations of ways an element can increase itself effects mediated by others. Is there a general mechanism to capture this type of interaction?