### Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem

N. P. van der Aa, H. G. ter Morsche, R. R.M. Mattheij
2007 The Electronic Journal of Linear Algebra
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
more » ... 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. http://math.technion.ac.il/iic/ela Note that D has the same partitioning as C. Now that all off-diagonal entries of C have been found, we use (3.5) to determine the diagonal entries to fill the coefficient matrix completely. Generalization. The cases discussed in the previous section provide us a way to generalize the theory of finding the first order derivatives of the eigenvectors for eigenvalues with the property that (some of) their derivatives up to the kth order are repeated, but the (k + 1)st order derivatives are distinct. This generalization will