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Karamardian Matrices: A Generalization of -Matrices
A real square matrix is called a -matrix if the linear complementarity
problem has a solution for all . This means that
for every vector there exists a vector such that and . A well known result of Karamardian states that if the problems
and for some have only the zero
solution, then is a -matrix. By relaxing the condition on and
imposing a condition on the solution vector in the two problems as above,
the authors introduce a new class of matrices called Karamardian matrices,
requiring that these two modified problems have only zero as a solution. In
this article, a systematic treatment of Karamardian matrices is undertaken.
Among other things, it is shown how Karamardian matrices have properties that
are analogous to those of -matrices. A subclass of a recently introduced
notion of -matrices is shown to possess the Karamardian property, and
for this reason we undertake a thorough study of -matrices and make
some fundamental contributions