Geodesic
Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem
The electronic journal of linear algebra, Tome 16 (2007), pp. 300-314.

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Summary: In many engineering applications, the physical quantities that have to be computed are obtained by solving a related eigenvalue problem. The matrix under consideration and thus its eigenvalues usually depend on some parameters. A natural question then is how sensitive the physical quantity is with respect to (some of) these parameters, i.e., how it behaves for small changes in the parameters. To find this sensitivity, eigenvalue and/or eigenvector derivatives with respect to those parameters need to be found. A method is provided to compute first order derivatives of the eigenvalues and eigenvectors for a general complex-valued, non-defective matrix.
Classification : 65F15
Keywords: eigenvalue derivatives, eigenvector derivatives, repeated eigenvalues 
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     title = {Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem},
     journal = {The electronic journal of linear algebra},
     pages = {300--314},
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     year = {2007},
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van der Aa, N.P.; ter Morsche, H.G.; Mattheij, R.R.M. Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem. The electronic journal of linear algebra, Tome 16 (2007), pp. 300-314. http://geodesic.mathdoc.fr/item/ELA_2007__16__a12/