Interval Drazin monotonicity of matrices (Q382133)

It is known that a real square matrix \(A\) is called monotone if \(Ax \geq 0\) implies \(x \geq 0\) for all \(x \in \mathbb{R}^n\). This concept has been generalized in many ways. In this paper, the authors introduce the notion of interval Drazin monotonicity and present a characterization of this concept and a new result for Drazin monotonicity. An \(n \times n\) real matrix \(A\) is said to be interval Drazin monotone if, for some interval \(J_2\) in \(\mathbb{R}^n\), there exists an interval \(J_1\) in \(\mathbb{R}^n\) such that \[ Ax \in J_2+N(A^k) \quad \text{and} \quad x \in R(A^k) \quad \text{imply} \quad x \in J_1, \] where \(A^k\) means that \(A\) is a matrix of index \(k\) (the least nonnegative integer such that \(\mathrm{rank}(A^{k+1})=\mathrm{rank}(A^k)\). The authors present a characterization of this concept in terms of interval boundedness. Specifically, they prove that a matrix \(A\) is interval Drazin monotone if and only if \(A^D\) is interval bounded. Then a sufficient condition for the nonnegativity of the Drazin inverse is provided by using the idea of nonnegative decomposition of matrices.