A real square matrix is monotone (in the sense of Collatz) if for all real vectors , implies , where is the element-wise order on .[1]

Properties

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A monotone matrix is nonsingular.[1]

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  .[1]

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

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The matrix   is monotone, with inverse  . 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  , whose inverse is  .

See also

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References

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