Monotone matrix

Not to be confused with monotonic matrix.

A real square matrix [math]\displaystyle{ A }[/math] is monotone (in the sense of Collatz) if for all real vectors [math]\displaystyle{ v }[/math], [math]\displaystyle{ Av \geq 0 }[/math] implies [math]\displaystyle{ v \geq 0 }[/math], where [math]\displaystyle{ \geq }[/math] is the element-wise order on [math]\displaystyle{ \mathbb{R}^n }[/math].

Properties

A monotone matrix is nonsingular.

Proof: Let [math]\displaystyle{ A }[/math] be a monotone matrix and assume there exists [math]\displaystyle{ x \ne 0 }[/math] with [math]\displaystyle{ Ax = 0 }[/math]. Then, by monotonicity, [math]\displaystyle{ x \geq 0 }[/math] and [math]\displaystyle{ -x \geq 0 }[/math], and hence [math]\displaystyle{ x = 0 }[/math]. [math]\displaystyle{ \square }[/math]

Let [math]\displaystyle{ A }[/math] be a real square matrix. [math]\displaystyle{ A }[/math] is monotone if and only if [math]\displaystyle{ A^{-1} \geq 0 }[/math].

Proof: Suppose [math]\displaystyle{ A }[/math] is monotone. Denote by [math]\displaystyle{ x }[/math] the [math]\displaystyle{ i }[/math]-th column of [math]\displaystyle{ A^{-1} }[/math]. Then, [math]\displaystyle{ Ax }[/math] is the [math]\displaystyle{ i }[/math]-th standard basis vector, and hence [math]\displaystyle{ x \geq 0 }[/math] by monotonicity. For the reverse direction, suppose [math]\displaystyle{ A }[/math] admits an inverse such that [math]\displaystyle{ A^{-1} \geq 0 }[/math]. Then, if [math]\displaystyle{ Ax \geq 0 }[/math], [math]\displaystyle{ x = A^{-1} Ax \geq A^{-1} 0 = 0 }[/math], and hence [math]\displaystyle{ A }[/math] is monotone. [math]\displaystyle{ \square }[/math]

Examples

The matrix [math]\displaystyle{ \left( \begin{smallmatrix} 1&-2\\ 0&1 \end{smallmatrix} \right) }[/math] is monotone, with inverse [math]\displaystyle{ \left( \begin{smallmatrix} 1&2\\ 0&1 \end{smallmatrix} \right) }[/math]. 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 [math]\displaystyle{ \left( \begin{smallmatrix} -1&3\\ 2&-4 \end{smallmatrix} \right) }[/math], whose inverse is [math]\displaystyle{ \left( \begin{smallmatrix} 2&3/2\\ 1&1/2 \end{smallmatrix} \right) }[/math].

See also

• M-matrix
• Weakly chained diagonally dominant matrix