AE Regularity of Interval Matrices

Abstract

Consider a linear system of equations with interval coefficients, and each interval coefficient is associated with either a universal or an existential quantifier. The AE solution set and AE solvability of the system is defined by ∀∃-quantification. Herein, we deal with the problem what properties must the coefficient matrix have in order that there is guaranteed an existence of an AE solution. Based on this motivation, we introduce a concept of AE regularity, which implies that the AE solution set is nonempty and the system is AE solvable for every right-hand side. We discuss characterization of AE regularity, and we also focus on various classes of matrices that are implicitly AE regular. Some of these classes are polynomially decidable, and therefore give an efficient way for checking AE regularity. We also state open problems related to computational complexity and characterization.

Generalization of real interval matrices to other fields

“…In [7] Hladík introduced the notion of strongly singular interval matrix. We generalize it to subset matrices over any field.…”

Interval matrices: realization of ranks by rational matrices

“…. , dt 2 ) is in Ker(G (α α αr,s),(c j ) ) (by (7) in case rk(G (α α αr,s),(c j ) ) ≥ 1 and by Remark A in case rk(G (α α αr,s),(c j ) ) = 0), hence it satisfies the equations ( 5) with cj instead of γ j ); then define ãi for i = 1, . .…”