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 βstableβ 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.References
- W. A. Coppel, Matrix quadratic equations, Bull. Austral. Math. Soc. 10 (1974), 377β401. MR 367381, DOI 10.1017/S0004972700041071
- John Daughtry, Isolated solutions of quadratic matrix equations, Linear Algebra Appl. 21 (1978), no.Β 1, 89β94. MR 485926, DOI 10.1016/0024-3795(87)90202-3 β, The inaccessible invariant subspaces of certain ${C_0}$ operators (preprint).
- R. G. Douglas and Carl Pearcy, On a topology for invariant subspaces, J. Functional Analysis 2 (1968), 323β341. MR 0233224, DOI 10.1016/0022-1236(68)90010-4
- J. Eisenfeld, Operator equations and nonlinear eigenparameter problems, J. Functional Analysis 12 (1973), 475β490. MR 0350474, DOI 10.1016/0022-1236(73)90007-4
- Kenneth Hoffman and Ray Kunze, Linear algebra, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0276251
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- Marvin Rosenblum, On the operator equation $BX-XA=Q$, Duke Math. J. 23 (1956), 263β269. MR 79235
Additional Information
- © Copyright 1979 American Mathematical Society
- 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