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

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 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.


15A24 Matrix equations and identities
41A05 Interpolation in approximation theory
47A57 Linear operator methods in interpolation, moment and extension problems


Zbl 0894.41001
Full Text: DOI


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