*(English)*Zbl 0987.90082

The paper deals with the SemiDefinite Linear Complementarity Problem (SDLCP$(L,{S}_{+}^{n})$: find a matrix $X\in {S}_{+}^{n}$ such that $Y=L\left(x\right)+Q\in {S}_{+}^{n}$ and and $\langle X,Y\rangle =0$, where ${S}^{n}$ (${S}_{+}^{n}$) denote the set of symmetric (positive semidefinite) matrices, $L:{S}^{n}\to {S}^{n}$ is a linear transformation, $Q\in {S}^{n}$ and $\langle X,Y\rangle $ denotes the trace of the matrix $XY$.

In this paper several LCP related concepts are extended to SDLCP. The well known LCP properties (${R}_{0}$ ${Q}_{0}$) 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 ${P}_{1}$ and ${P}_{2}$) -properties). The authors consider in particular these properties for Lyapunov transformation ${L}_{A}=AX+X{A}^{T}\xb7$ It is shown the equivalence between some of these properties for ${L}_{A}$, and it is also proven that the P-property (the Q-property) is equivalent to $A$ being a positive stable (i.e., real parts of eigenvalues of $A$ are positive). As a special case, a theorem of Lyapunov is deduced.

As shown by a counterexample, the P-property of $L$ 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 $L$. It is proven that ${L}_{A}$ has the GUS property iff A is positive stable (i.e., and positive semidefinite).