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

DOI:
https://doi.org/10.1090/S0002-9939-1979-0521866-X

MathSciNet review:
521866

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Abstract: The authors determine which solutions *K* to the quadratic matrix equation 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:
https://doi.org/10.1090/S0002-9939-1979-0521866-X

Keywords:
Perturbation theory,
stable invariant subspaces,
quadratic matrix equations

Article copyright:
© Copyright 1979
American Mathematical Society