Eigenvalue and eigenvector sensitivities in linear systems theory†

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Abstract
Simplo and explicit derivations arc given of expressions for the eigenvalue and eigenvector sensitivity coefficients for the fundamental eigenproblem associated with tho behaviour of linear systems governed by equations of the form x = Ax. Such expressions relating changes in the eigenvalues and eigenvectors of tho matrix A to changes in A have been given by numerous authors since tho early work of Jaeobi (1846). These expressions have been given in various forms and from different points of view, and have also heen derived by a number of different methods. Thus, the sensitivity cigenproblem has been viewed as a problem in numerical analysis (Faddeev and Faddecva 1963, Wilkinson 1965), as a problem in perturbation theory (Bellman 1960, Wilkinson 1965), and as a problem in linear systems theory (Laughton HI64, Mann and Marshall 1965, Mann et al. 1965, Van Ness et al. 1965, Rosenbrock 1965, Morgan 1966 a, b, Nicholson 1967 a, b). It is hoped that this paper may help to clarify n.ny obscurities which may have resulted from tho existence of these different but logically equivalent solutions of the same problem: the paper by Laughton (1964) is commented upon in some detail.

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