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

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

The matrix $M=\left({\begin{array}{rr}1&-2\\&1\end{array}}\right)$ is monotone, with inverse $M^{-1}={\begin{pmatrix}1&2\\&1\end{pmatrix}}$ . In fact, $M$ is an M-matrix (an M-matrix is a monotone L-matrix).

Note, however, that not all monotone matrices are M-matrices. An example is $N={\begin{pmatrix}-1&3\\2&-4\end{pmatrix}}$ , whose inverse is $N^{-1}={\begin{pmatrix}2&3/2\\1&1/2\end{pmatrix}}$ .

References

1 2 3 Mangasarian, O. L. (1968). "Characterizations of Real Matrices of Monotone Kind". SIAM Review. 10 (4): 439–441. doi:10.1137/1010095. ISSN 0036-1445.

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Monotone matrix

Properties

Examples

See also

References