Forty necessary and sufficient conditions for regularity of interval matrices: A survey

Abstract

This is a survey of forty necessary and sufficient conditions for regularity of interval matrices published in various papers over the last thirty-five years. Afull list of references to the sources of all the conditions is given, and they are commented on in detail. 1. Introduction. During the last thirty-five years (1973-2008), considerable in- terest has been dedicated to the problemof regularity of interval m atrices. It has resulted in formulations of altogether forty necessary and sufficient conditions that constitute the subject matter of this survey paper. By definition, a square interval matrix A is called regular if each A ∈ A is nonsingular, and it is said to be singular otherwise (i.e., if it contains a singular matrix). It is the purpose of this paper to show that this property can be reformulated in surprisingly many surprisingly various ways. In the main Theorem 4.1 we show that regularity of interval matrices can be characterized in terms of determinants (Theorem 4.1, condition (xxxii)), matrix inverses (xxx), linear equations (xxv), absolute value equations (v), absolute value inequalities (ii), matrix equations (xxiv), solvability in each orthant (xvi), inclusions (xxxvii), set properties (xxxvi), real spectral radius (xxxiv), P -matrices (xxix) and edge nonsingularity (xli). We do not include the proof of mutual equivalence of all the conditions since that would make for a very lengthy