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 $\mathbb{R}^n$ .{{cite journal|last1=Mangasarian|first1=O. L.|title=Characterizations of Real Matrices of Monotone Kind|journal=SIAM Review|volume=10|issue=4|year=1968|pages=439–441|issn=0036-1445|doi=10.1137/1010095|url=https://minds.wisconsin.edu/bitstream/handle/1793/57482/TR15.pdf?sequence=1}}

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

References

Category:Matrices (mathematics)

Monotone matrix

Properties

Examples

See also

References