A nonlinear matrix equation connected to interpolation theory

In this paper we study the matrix equation where Q is an n×n positive definite matrix, C is an mn×mn positive semidefinite matrix, A is an arbitrary mn×n matrix and is the m×m block diagonal matrix with on each diagonal entry the n×n matrix X. We are interested in the existence and uniqueness of solutions which are contained in a certain subset of the set of the positive definite matrices, under the condition that . These solutions play a role in an optimal interpolation theory problem [Interpolation Theory and Its Applications, Mathematics and Its Applications, vol. 428, Kluwer Academic, Dordrecht, 1997 (Chapter 7)].