Monotone matrix


A real square matrix is monotone if for all real vectors, implies, where is the element-wise order on.

Properties

A monotone matrix is nonsingular.
Proof: Let be a monotone matrix and assume there exists with. Then, by monotonicity, and, and hence.
Let be a real square matrix. is monotone if and only if.
Proof: Suppose is monotone. Denote by the -th column of. Then, is the -th standard basis vector, and hence by monotonicity. For the reverse direction, suppose admits an inverse such that. Then, if,, and hence is monotone.

Examples

The matrix is monotone, with inverse.
In fact, this matrix is an M-matrix.
Note, however, that not all monotone matrices are M-matrices. An example is, whose inverse is.