Mathematical Programming 35 (1986) 17-31 North-Holland A FINITE CHARACTERIZATION OF K-MATRICES IN DIMENSIONS LESS THAN FOUR John T. FREDRICKSEN* Amdahl Corporation, 1250 East Arques Avenue, Sunnyvale, CA 94088-3470, USA Layne T. WATSON* Department of Computer Science, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Katta G. MURTY** Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, M148109, USA The class of real n × n matrices M, known as K-matrices, for which the linear complementarity problem w - Mz = q, w >1 O, z >1 O, WTZ = 0 has a solution whenever w - Mz = q, w >IO, z >1 0 has a solution is characterized for dimensions n <4. The characterization is finite and 'practical'. Several necessary conditions, sufficientconditions, and counterexamples pertaining to K-matrices are also given. A finite characterization of completely K-matrices (K-matrices all of whose principal submatrices are also K-matrices) is proved for dimensions <4. Key words: Linear complementarity problem, K-matrix, Q0-matrix, finite characterization, Q-matrix. I. Introduction Let E" be the n-dimensional Euclidean space and let E "×" be the set of real nxn matrices. For M~E "x", Mo denotes the (i,j) entry in M, and for /, Jc {1,..., n}, MI. is the submatrix of M consisting of the rows indexed by I; and M.j consists of the columns indexed by J. The jth column of M is denoted either Mj or M.j, the ith row of M is denoted by Mi.. Given a matrix M ~ E"×" and vector q c E", the linear complementarity problem, denoted by (q, M), is to find vectors w, z~ E" such that w-Mz=q, (LCP) w>lO, z>lO, wTz=O. This problem arises in such diverse areas as economics, game theory, linear program- ming, mechanics, lubrication, numerical analysis, and nonlinear optimization. Gen- erally in a particular application area the matrix M has a special structure (e.g., * Partially supported by NSF Grant MCS-8207217. ** Research partially supported by NSF Grant No. ECS-8401081. 17