A nonlinear matrix equation connected to interpolation theory.
The authors study the matrix equation \(X=Q+A^*(\widehat{X}-C)^{-1}A\), where \(Q\) is an \(n\times n\) positive definite matrix, \(C\) is \(mn\times mn\) positive semidefinite, \(A\) is \(mn\times n\) (arbitrary) and \(\widehat{X}\) is \(mn\times mn\), block diagonal, with \(m\) diagonal blocks equal to \(X\). The authors impose the condition \(C<\widehat{Q}\) (i.e. the matrix \(\widehat{Q}-C\) is positive definite) and prove the existence and uniqueness of the solution in a certain class of positive definite matrices. These solutions are important in a problem from optimal interpolation theory, see \textit{L. A. Sakhnovich} [Interpolation theory and its Applications. (Mathematics and Its applications. (Dordrecht). 428 Dordrecht: Kluwer Academic Publishers.) (1997; Zbl 0894.41001), Chapter 7], where existence and uniqueness of the solutions is conjectured.
scientific article; zbMATH DE number 2056691