Qualitative convergence of matrices (Q1019629)

A complex square matrix \(A\) is stable if all its eigenvalues lie in the open left half-plane of \(\mathbb{C}\), and convergent if they lie within the open unit disk. Qualitative analysis, where the entries of \(A\) are not specified, arose from the study of stability problems in economics and ecology. In the continuous time case, only the signs of the entries of the real matrix \(A\) (its sign pattern) are given. In the discrete time case, what is given for each entry of \(A\) is whether its modulus is \(0\) or \(1\) or in \((0,1)\) or greater than \(1\) (the modulus pattern of \(A\)). Such qualitative information about \(A\) can be used to determine the stability or convergence of \(A\). In this paper the author introduces several types of potential convergence, the basic definition of which is: a modulus pattern \(Z\) is potentially convergent if there exists a convergent matrix with the modulus pattern \(Z\). All of these different types are characterized for matrices of size 2. Potential absolute convergence is shown to be equivalent to potential diagonal convergence and their complete characterization for size \(n\) is given.