Monotone matrix
# 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 $x\neq 0$ 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 $\left({\begin{smallmatrix}2&3/2\\1&1/2\end{smallmatrix}}\right)$.

