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 $\mathbb {R} ^{n}$ .^[1]

Properties[edit]

A monotone matrix is nonsingular.^[1]

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 $A^{-1}\geq 0$ .^[1]

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[edit]

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 $\left({\begin{smallmatrix}2&3/2\\1&1/2\end{smallmatrix}}\right)$ .