On the nonlinear matrix equation \(X+A^*{\mathcal F}(X)A=Q\): solutions and perturbation theory
Denote by \({\mathcal M}(n)\), \({\mathcal P}(n)\) respectively the sets of all \(n\times n\) and of all \(n\times n\) Hermitian positive semidefinite matrices. The nonlinear matrix equation \(X+A^*{\mathcal F}(X)A=Q\) is considered where \({\mathcal F}:{\mathcal P}(n)\rightarrow {\mathcal M}(n)\), \(Q\) is Hermitian positive definite and \(A\in {\mathcal M}(n)\). Sufficient conditions for the existence and uniqueness of a positive semidefinite solution are derived. Particular attention is paid to the cases when \({\mathcal F}:{\mathcal P}(n)\rightarrow \pm {\mathcal P}(n)\) is monotone or antimonotone, i.e. when \(X\leq Y\) implies respectively \({\mathcal F}(X)\leq {\mathcal F}(Y)\) or \({\mathcal F}(X)\geq {\mathcal F}(Y)\). (The inequality \(X\leq Y\) means that \(Y-X\) is positive semidefinite.) Conditions are given for the continuous dependence of the solution on \(A\) and \(Q\). The perturbed equation is also considered.
scientific article; zbMATH DE number 1758606