On maximum principles for monotone matrices
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)].
scientific article; zbMATH DE number 3941716
scientific article