Complementarity Problems And Positive Definite Matrices - Bhimashankaram, P.; Parthasarathy, T.; Murthy, A. L. N.; Murthy, G. S. R. (Algorithmic Operations Research, 2012)
The class of positive definite and positive semidefinite matrices is one of the most frequently encountered matrix classes both in theory and practice. In statistics, these matrices appear mostly with symmetry. However, in complementarity problems generally symmetry in not necessarily an accompanying feature. Linear complementarity problems defined by positive semidefinite matrices have some interesting properties such as the solution sets are convex and can be processed by Lemke’s algorithm as well as Graves’ principal pivoting algorithm. It is known that the principal pivotal transforms (PPTs) (defined in the context of linear complementarity problem) of positive semidefinite matrices are all positive semidefinite. In this article, we introduce the concept of generalized PPTs and show that the generalized PPTs of a positive semidefinite matrix are also positive semidefinite. One of the important characterizations of P -matrices (that is, the matrices with all principle minors positive) is that the corresponding linear complementarity problems have unique solutions. In this article, we introduce a linear transformation and characterize positive definite matrices as the matrices with corresponding semidefinite linear complementarity problem having unique solutions. Furthermore, we present some simplification procedure in solving a particular type of semidefinite linear complementarity problems involving positive definite matrices.