A real square matrix A is monotone (in the sense of Collatz) if for all real vectors v , A v ≥ 0 implies v ≥ 0 , where ≥ is the element-wise order on R n .
A monotone matrix is nonsingular.
Proof: Let A be a monotone matrix and assume there exists x ≠ 0 with A x = 0 . Then, by monotonicity, x ≥ 0 and − x ≥ 0 , and hence x = 0 . ◻
Let A be a real square matrix. A is monotone if and only if A − 1 ≥ 0 .
Proof: Suppose A is monotone. Denote by x the i -th column of A − 1 . Then, A x is the i -th standard basis vector, and hence x ≥ 0 by monotonicity. For the reverse direction, suppose A admits an inverse such that A − 1 ≥ 0 . Then, if A x ≥ 0 , x = A − 1 A x ≥ A − 1 0 = 0 , and hence A is monotone. ◻
The matrix ( 1 − 2 0 1 ) is monotone, with inverse ( 1 2 0 1 ) . 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 ( − 1 3 2 − 4 ) , whose inverse is ( 2 3 / 2 1 1 / 2 ) .
