New characterizations of row sufficient matrices

In dealing with a linear complementarity problem, much depends on knowing that the matrix, through which the particular LCP is defined, belongs to a suitable matrix class. Two such classes are SU – the so-called sufficient matrices – and L which were introduced in [R.W. Cottle, J.-S. Pang, V. Venkateswaran, Sufficient matrices and the linear complementarity problem, Linear Algebra Appl. 114/115 (1989) 231–249; B.C. Eaves, The linear complementarity problem, Manage. Sci. 17 (1971) 612–634], respectively. In an earlier article [I. Adler, R.W. Cottle, S. Verma, Sufficient matrices belong to L, Math. Prog. 106 (2006) 391–401], the authors proved that SU is a subclass of L. By definition, the class SU is the intersection of two distinct classes: RSU, the row sufficient matrices, and CSU, the column sufficient matrices. In the present work, we strengthen the aforementioned inclusion by showing that all row sufficient matrices belong to L. Using what we call “structural properties” of certain matrix classes, we add to the existing characterizations of RSU in [R.W. Cottle, S.-M. Guu, Two characterizations of sufficient matrices, Linear Algebra Appl. 170 (1992) 65–74; S.-M. Guu, R.W. Cottle, On a subclass of P0, Linear Algebra Appl. 223/224 (1995) 325–335; H. Väliaho, Criteria for sufficient matrices, Linear Algebra Appl. 233 (1996) 109–129]. This line of inquiry was inspired by asking: what must be true of a row sufficient L-matrix? We establish three new characterizations of RSU in terms of the matrix classes L, E0, and Q0 and the structural properties of sign-change invariance, completeness, and fullness. The new characterizations of RSU provide new characterizations of SU by adjoining a fourth structural property we call reflectiveness.