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Monotone matrix

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(Redirected from Inverse-positive matrix) Not to be confused with monotonic matrix.
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For other uses, see SMAWK algorithm and Monge array.

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

Properties

A monotone matrix is nonsingular.

Proof: Let A {\displaystyle A} be a monotone matrix and assume there exists x ≠ 0 {\displaystyle x\neq 0} with A x = 0 {\displaystyle Ax=0} . Then, by monotonicity, x ≥ 0 {\displaystyle x\geq 0} and − x ≥ 0 {\displaystyle -x\geq 0} , and hence x = 0 {\displaystyle x=0} . ◻ {\displaystyle \square }

Let A {\displaystyle A} be a real square matrix. A {\displaystyle A} is monotone if and only if A − 1 ≥ 0 {\displaystyle A^{-1}\geq 0} .

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

Examples

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

See also

References

  1. ^ 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.
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