About Hölder condition numbers and the stratification diagram for defective eigenvalues

In this paper, we look at a particular case of application which is Holder continuous, namely the map from a matrix A to one of its eigenvalues λ, when it is multiple defective. Two asymptotic Holder condition numbers are considered : one (resp. the other) is associated with a generalization of the Frechet (resp. Gateaux) derivative [9].We illustrate on a fully developed 3 × 3 example, why these asymptotic condition numbers may not be appropriate to analyze eigencomputations performed in finite precision. We present the complementary view point of differential geometry, developed by A. Ilahi in [12], which is based on the stratification diagram in C3 × 3. The distance to the stratum indicates when this complementary viewpoint is required.