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

Ran, André C. M.; Reurings, Martine C. B.

Linear Algebra Appl. 379, 289-302 (2004).

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.

Reviewer: Vladimir P. Kostov (Nice)

MSC:

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

Keywords:

nonlinear matrix equations; interpolation theory; generalized Riccati equation; positive definite matrices

Citations:

Zbl 0894.41001

References:

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