Geodesic
A counterexample for characterizing an invariant subspace of a matrix
Electronic transactions on numerical analysis, Tome 31 (2008), pp. 295-305
As an alternative to Newton's method for computing a simple eigenvalue and corresponding eigenvectors of a nonnormal matrix in a stable way, an approach based on singularity theory has been proposed by Schwetlick/L$\ddot $osche [Z. Angew. Math. Mech., 80 (2000), pp. 9-25]. In this paper, by constructing a counterexample with a singular linear block operator, it is shown that a straightforward extension of this technique to the computation of invariant subspaces of dimension p > 1 will not work, in general. Finding this counterexample required a detailed study of the linear block operator.
Classification : 65F15
Keywords: eigenvalue problem, simple invariant subspace, block Newton method, block Rayleigh quotient iteration 
@article{ETNA_2008__31__a5,
     author = {Schwetlick,  Hubert and Schreiber,  Kathrin},
     title = {A counterexample for characterizing an invariant subspace of a matrix},
     journal = {Electronic transactions on numerical analysis},
     pages = {295--305},
     year = {2008},
     volume = {31},
     zbl = {1189.65071},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2008__31__a5/}
}
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Schwetlick,  Hubert; Schreiber,  Kathrin. A counterexample for characterizing an invariant subspace of a matrix. Electronic transactions on numerical analysis, Tome 31 (2008), pp. 295-305. http://geodesic.mathdoc.fr/item/ETNA_2008__31__a5/