Qualitative convergence of matrices

In this paper, we study potential convergence of modulus patterns. A modulus pattern Z is convergent if all complex matrices in Q(Z) (i.e. all matrices with modulus pattern Z) are convergent. A modulus pattern is potentially (absolutely) convergent if there exists a (nonnegative) convergent matrix in Q(Z). We also introduce types of potential convergence that correspond to diagonal and D-convergence, studied in [E. Kaszkurewicz, A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhauser, 2000]. Convergent modulus patterns have been completely characterized by Kaszkurewicz and Bhaya [E. Kaszkurewicz, A. Bhaya, Qualitative stability of discrete-time systems, Linear Algebra Appl. 117 (1989) 65–71]. This paper presents some techniques that can be used to establish potential convergence. Potential absolute convergence and potential diagonal convergence are shown to be equivalent, and their complete characterization for n×n modulus patterns is given. Complete characterizations of all introduced types of potential convergence for 2×2 modulus patterns are also presented.