## On maximum principles for monotone matrices.(English)Zbl 0587.15014

The (n,n)-matrix A satisfies the maximum principle if $$Ay=f$$ and $$f\geq 0$$ implies $$y\geq 0$$, and moreover $$\max \{y_ i|$$ $$i\in N\}=\max \{y_ i| i\in N^+(f)\}$$ with $$N=\{1,...,n\}$$ and $$N^+(f)=\{j\in N| f_ j>0\}$$. Define $$A^{(+)}=a_{ij}$$ for $$i\neq j$$ and $$a_{ij}>0$$ and 0 elsewhere. The following holds. Let A be nonsingular with nonnegative inverse and such that $$A-A^{(+)}$$ is either nonsingular, or singular and irreducible. Moreover, let $$A-A^{(+)}$$ have nonnegative row sums. Then A satisfies the maximum principle. Analogous properties of the solutions to linear equations with an irreducible M-matrix are studied in connection with the open Leontief input-output model [the reviewer, ibid. 26, 175-201 (1979; Zbl 0409.90027)].
Reviewer: G.Sierksma

### MSC:

 15B48 Positive matrices and their generalizations; cones of matrices 93D25 Input-output approaches in control theory 15A09 Theory of matrix inversion and generalized inverses

Zbl 0409.90027
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### References:

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