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 \(A\geq 0\). 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


15A24 Matrix equations and identities
15B48 Positive matrices and their generalizations; cones of matrices
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