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As an alternative to Newton's method an approach based on singularity
theory can be exploited for computing a simple eigenvalue and corresponding
eigenvectors of a nonnormal matrix $A$ in a stable way, see
[Schwetlick/L\"osche, 2000].
In the paper it is shown by constructing a counterexample with an singular
singularity system, that a straightforward extension of this technique
to the computation of invariant subspaces of dimension $p>1$
will not work, in general. The finding of this counterexample required a
detailed study of the linear block singularity operator.