Monotone matrix

A real square matrix formula_1 is monotone (in the sense of Collatz) if for all real vectors formula_2, formula_3 implies formula_4, where formula_5 is the element-wise order on formula_6.
Properties.
A monotone matrix is nonsingular.
"Proof": Let formula_1 be a monotone matrix and assume there exists formula_8 with formula_9. Then, by monotonicity, formula_10 and formula_11, and hence formula_12. formula_13
Let formula_1 be a real square matrix. formula_1 is monotone if and only if formula_16.
"Proof": Suppose formula_1 is monotone. Denote by formula_18 the formula_19-th column of formula_20. Then, formula_21 is the formula_19-th standard basis vector, and hence formula_10 by monotonicity. For the reverse direction, suppose formula_1 admits an inverse such that formula_16. Then, if formula_26, formula_27, and hence formula_1 is monotone. formula_13
Examples.
The matrix formula_30 is monotone, with inverse formula_31.
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 formula_32, whose inverse is formula_33.