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A nonlinear matrix equation connected to interpolation theory. (English) Zbl 1039.15007

The authors study the matrix equation X=Q+A * (X ^-C) -1 A, where Q is an n×n positive definite matrix, C is mn×mn positive semidefinite, A is mn×n (arbitrary) and X ^ is mn×mn, block diagonal, with m diagonal blocks equal to X. The authors impose the condition C<Q ^ (i.e. the matrix 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 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.


MSC:
15A24Matrix equations and identities
41A05Interpolation (approximations and expansions)
47A57Operator methods in interpolation, moment and extension problems