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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The stable solutions of quadratic matrix equations


Authors: Stephen Campbell and John Daughtry
Journal: Proc. Amer. Math. Soc. 74 (1979), 19-23
MSC: Primary 15A24; Secondary 47A55
MathSciNet review: 521866
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Abstract: The authors determine which solutions K to the quadratic matrix equation $ XBX + XA - DX - C = 0$ are ``stable'' in the sense that all small changes in the coefficients of the equation produce equations some of whose solutions are close to K (in the metric determined by the operator norm). Our main result is that a solution is stable if and only if it is an isolated solution. (The isolated solutions already have a simple characterization in terms of the coefficient matrices.) It follows that each equation has only finitely many stable solutions.

Equivalently, we identify the stable invariant subspaces for an operator T on a finite-dimensional space as the isolated invariant subspaces.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0521866-X
PII: S 0002-9939(1979)0521866-X
Keywords: Perturbation theory, stable invariant subspaces, quadratic matrix equations
Article copyright: © Copyright 1979 American Mathematical Society