Page 157 - Textos de Matemática Vol. 39

ON THE STABILITY OF INVARIANT SUBSPACES OF SQUARE MATRICES
ION ZABALLA
Dedicated to Eduardo Marques de Sa´ on the occasion of his 60th birthday.
Abstract. A new proof of a classical result characterizing the invariant sub- spaces of endomorphisms which are stable is provided. Only matrix tools are used and it is strongly based on some perturbation results on eigenvalues and subspaces.
1. Introduction
For a square complex matrix A a classical result establishes that if A is derogatory then the only invariant subspaces which are stable are {0} and the radical subspaces; and if A is nonderogatory all invariant subspaces are stable (see [3] and the references therein). Recall that a subspace S is A-invariant if AS ⊂ S; and it is said to be stable if ∀ε > 0 ∃ δ > 0 such that if ∥A − A′∥ < δ then A′ has an invariant subspace S′ satisfying Θ(S, S′) < ε. Here Θ(·, ·) stands for the gap metric between subspaces (see [3]). The fact that for this definition to make sense matrices A and A′ must be close enough to each other seems to indicate that the characterization of the Jordan canonical forms of the nearby matrices could play a role in the solution of this problem. This manuscript is the result of a search in such a direction. It strongly relies on some fundamental results about perturbation theory of eigenvalues and invariant subspaces for which our basic reference is [2].
Along this paper, C will denote the field of complex numbers although some algebraic properties will hold for matrices with elements in an arbitrary field that we will denote by F. If X ∈ Fp×q the symbol < X > will mean the subspace of Fp spanned by the columns of X; i.e. ImX. Cn and Cp×q will be provided with the topology derived from any norm, ∥ · ∥. In some specific
2000 Mathematics Subject Classification. 15A60, 15A42, 15A18.
Key words and phrases. Invariant subspaces, endomorphisms, simple decomposition, nilpontent, nonderogatory, root subspces.
The work was supported by the MEC Pro ject MTM2004-06389-CO2-01 and the UPV/EHU Project GUI-2004.
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