Monotonicity of matrices with positive main diagonal (Q1095990)

One definition of monotonicity of a real \(n\times n\) matrix \(A=\{a_{ij}\}\) requires that it be invertible and \(A^{-1}\geq 0\) [cf. \textit{L. Collatz}, Arch. Math. 3, 366-376 (1952; Zbl 0048.098)]. The author considers necessary and sufficient conditions for the monotonicity of a matrix A which a priori satisfies \(a_{ii}>0\), \(i=1,...,n\); \(a_{ij}\leq 0\), \(i\neq j\). The sufficient condition of diagonal dominance, attributed to Collatz, is well-known; and a necessary and sufficient condition is due to Varga. Several of the author's necessary conditions are obtained under the additional prior assumption of irreducibility of A. His sufficient conditions are derived from \textit{R. S. Varga}'s theorem [Matrix Iterative Analysis (1963; Zbl 0133.086) p. 84]; as well as from the localization theorems of Brauer and of Ostrowski.