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Zbl 0702.15009
Anderson, W.N.jun.; Morley, T.D.; Trapp, G.E.
Positive solutions to $X=A-BX\sp{-1}B\sp*$. (English)
[J] Linear Algebra Appl. 134, 53-62 (1990). ISSN 0024-3795

The authors study the positive (semidefinite) solutions to the matrix equation $X=A-BX\sp{-1}B\sp*$ under the assumption that $A\ge 0$. 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.
[M.de la Sen]

MSC 2000:: *15A24 Matrix equations
15A48 Positive matrices and their generalizations

Keywords: bounded operators; matrix equation; positive solutions; Schur complement; Hilbert space; inner product

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Zentralblatt MATH Berlin [Germany]