## A counterexample for characterizing an invariant subspace of a matrix

Hubert Schwetlick and Kathrin Schreiber

### Abstract

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/Lösche [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.

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### Key words

Eigenvalue problem, simple invariant subspace, block Newton method, block Rayleigh quotient iteration.

65F15.

### Links to the cited ETNA articles

 [8] Vol. 7 (1998), pp. 56-74 Jean-Luc Fattebert: A block Rayleigh quotient iteration with local quadratic convergence

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