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Positive solutions to X=A-BX -1 B * . (English) Zbl 0702.15009
The authors study the positive (semidefinite) solutions to the matrix equation X=A-BX -1 B * under the assumption that A0. It is shown that positive solutions exist if and only if a certain block tridiagonal operator is positive, in which case the solution is given by the generalized Schur complement of that operator. The Schur complement is considered to act on a proper subspace of a finite or infinite dimensional Hilbert space with inner product.
Reviewer: M.de la Sen
MSC:
15A24Matrix equations and identities
15A48Positive matrices and their generalizations (MSC2000)