Monotone matrix

A real square matrix $$A$$ is monotone (in the sense of Collatz) if for all real vectors $$v$$, $$Av \geq 0$$ implies $$v \geq 0$$, where $$\geq$$ is the element-wise order on $$\mathbb{R}^n$$.

Properties
A monotone matrix is nonsingular.

Proof: Let $$A$$ be a monotone matrix and assume there exists $$x \ne 0$$ with $$Ax = 0$$. Then, by monotonicity, $$x \geq 0$$ and $$-x \geq 0$$, and hence $$x = 0$$. $$\square$$

Let $$A$$ be a real square matrix. $$A$$ is monotone if and only if $$A^{-1} \geq 0$$.

Proof: Suppose $$A$$ is monotone. Denote by $$x$$ the $$i$$-th column of $$A^{-1}$$. Then, $$Ax$$ is the $$i$$-th standard basis vector, and hence $$x \geq 0$$ by monotonicity. For the reverse direction, suppose $$A$$ admits an inverse such that $$A^{-1} \geq 0$$. Then, if $$Ax \geq 0$$, $$x = A^{-1} Ax \geq A^{-1} 0 = 0$$, and hence $$A$$ is monotone. $$\square$$

Examples
The matrix $$\left( \begin{smallmatrix} 1&-2\\ 0&1 \end{smallmatrix} \right)$$ is monotone, with inverse $$\left( \begin{smallmatrix} 1&2\\ 0&1 \end{smallmatrix} \right)$$. In fact, this matrix is an M-matrix (i.e., a monotone L-matrix).

Note, however, that not all monotone matrices are M-matrices. An example is $$\left( \begin{smallmatrix} -1&3\\ 2&-4 \end{smallmatrix} \right)$$, whose inverse is $$\left( \begin{smallmatrix} 2&3/2\\ 1&1/2 \end{smallmatrix} \right)$$.