The paper deals with the SemiDefinite Linear Complementarity Problem (SDLCP: find a matrix such that and and , where () denote the set of symmetric (positive semidefinite) matrices, is a linear transformation, and denotes the trace of the matrix .
In this paper several LCP related concepts are extended to SDLCP. The well known LCP properties ( ) are extended to linear transformations, as well as semi-monotone , strictly semi-monotone, column sufficiency, cross commutative, and variations of the P-property (so called and ) -properties). The authors consider in particular these properties for Lyapunov transformation It is shown the equivalence between some of these properties for , and it is also proven that the P-property (the Q-property) is equivalent to being a positive stable (i.e., real parts of eigenvalues of are positive). As a special case, a theorem of Lyapunov is deduced.
As shown by a counterexample, the P-property of does not imply the uniqueness of a solutions in the SDLCP. So, in order to address the unique solvability of SDLCP the author introduces the globally uniquely solvable property (GUS-property) of a linear transformation . It is proven that has the GUS property iff A is positive stable (i.e., and positive semidefinite).