Given the following matrix equation: $$AX=b$$ with:
- $A$ a (n,n) invertible matrix ;
- $X$ and $b$ vectors of size $n$ with all elements of $b$ positive or null,
what is the condition on $A$ to have $X$ to have all his elements positive or null.
I wonder if the response is so trivial... (i also feel shameful to ask this question :) )
If we want $x$ to be nonnegative componentwise regardless of $b$, we need $A^{-1}$ to be nonnegative componentwise. Suppose not, then suppose $A^{-1}$ has a negative entry in column $j$, then we can let $b=e_j$ where $e_j$ is the $j$-th unit standard basis vector to construct a counter example.
You are looking for monotone matrix, matrices where if $Av \ge 0$ then $v \ge 0$ where the inequality is defined componentwise.
A subclass of monotone matrix is the M-matrix. A Z-matrix is a matrix where the off-diagonal entry is non-positive. An $M$-matrix, $A$, is a $Z$-matrix where it can be written as $A=sI-B$ where $s$ is at least as large as the maximum of the moduli of the engenvalues of $B$ where $B$ is nonnegative componentwise. Equivalence condition of $M$-matrices can be found on the wikipedia page.