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**On semidefinite linear complementarity problems.**
*(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 \(Y=L(x)+Q \in S^n_+\) and and \(\langle X,Y\rangle=0\), where \(S^n\) (\(S^n_+\)) denote the set of symmetric (positive semidefinite) matrices, \(L: S^n \rightarrow S^n\) 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 \(L_A=AX+XA^T.\) 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).

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 \(L_A=AX+XA^T.\) 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).

Reviewer: S.M.Allende-Alonso (Ciudad Habana)