Characterization of \(Q\)-property for multiplicative transformations in semidefinite linear complementarity problems
Let \(V\overset.{}={\mathcal S}^{n\times n}\) be the vector space of real symmetric matrices of order \(n\) and \(\Sigma\) be the set of all posiive semidefinite matrices in \(V\). If \(X\in\Sigma\), we will use the notation \(X\succeq 0\). Suppose that \(L: V\to V\) is a linear transformation. Given an element \(Q\in V\), the semidefinite linear complementarity problem \(\text{SDLCP}(L, Q)\) is to find a matrix \(X\in V\) such that \[ X\succeq 0,\quad Y:= L(X)+ Q\succeq 0\quad\text{and}\quad XY= 0. \] Let \(A\) be a square matrix of order \(n\), the multiplicative transformation \(M_A: V\to V\) is defined by \(M_A(X)= AXA^\top\), and it is said to have the \(Q\)-property is \(\text{SDLCP}(M_A, Q)\) has solution for all \(Q\in V\). The following result is proved. Given \(A\in\mathbb{R}^{n\times n}\) the following are equivalent (a) \(A+ A^\top\) is either positive definite or negative definite; (b) For all \(Q\in V\), \(\text{SDLCP}(M_A, Q)\) has a unique solution; (c) \(\text{SDLCP}(M_A, 0)\) has a unique solution; (d) \(M_A\) has the \(Q\)-property.
scientific article; zbMATH DE number 5702818