Monotone matrix

Not to be confused with monotonic matrix.

A real square matrix $A$ is monotone (in the sense of Collatz) if for all real vectors $v$, $Av\ge 0$ implies $v\ge 0$, where $\ge$ 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, $x\ge 0$ and $-x\ge 0$, and hence $x=0$. $◻$

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

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 $x\ge 0$ by monotonicity. For the reverse direction, suppose $A$ admits an inverse such that $A^{-1}\ge 0$. Then, if $Ax\ge 0$, $x=A^{-1}Ax\ge A^{-1}0=0$, and hence $A$ is monotone. $◻$

Examples

The matrix $\left(\begin{array}{cc}1& -2\\ 0& 1\end{array}\right)$ is monotone, with inverse $\left(\begin{array}{cc}1& 2\\ 0& 1\end{array}\right)$. 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 $\left(\begin{array}{cc}-1& 3\\ 2& -4\end{array}\right)$, whose inverse is $\left(\begin{array}{cc}2& 3/2\\ 1& 1/2\end{array}\right)$.

See also

• M-matrix
• Weakly chained diagonally dominant matrix