Abstract.
Given a linear transformation L:?n→?n and a matrix Q∈?n, where ?n is the space of all symmetric real n×n matrices, we consider the semidefinite linear complementarity problem SDLCP(L,?n +,Q) over the cone ?n + of symmetric n×n positive semidefinite matrices. For such problems, we introduce the P-property and its variants, Q- and GUS-properties. For a matrix A∈R n×n, we consider the linear transformation L A :?n→?n defined by L A (X):=AX+XA T and show that the P- and Q-properties for L A are equivalent to A being positive stable, i.e., real parts of eigenvalues of A are positive. As a special case of this equivalence, we deduce a theorem of Lyapunov.
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Received: March 1999 / Accepted: November 1999¶Published online April 20, 2000
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Gowda, M., Song, Y. On semidefinite linear complementarity problems. Math. Program. 88, 575–587 (2000). https://doi.org/10.1007/PL00011387
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DOI: https://doi.org/10.1007/PL00011387