The stable solutions of quadratic matrix equations

Campbell, Stephen; Daughtry, John

doi:10.1090/S0002-9939-1979-0521866-X

The stable solutions of quadratic matrix equations
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by Stephen Campbell and John Daughtry PDF

Proc. Amer. Math. Soc. 74 (1979), 19-23 Request permission

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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