Page 158 - Textos de Matemática Vol. 39

148 ION ZABALLA
computations we will use the l1 norm for vector and matrices:
∥ x ∥1= ∥ X ∥1=
|xi|, p q i=1 j=1
x ∈ Cn.
|xij|, X = [xij] ∈ Cp×q.
n i=1
For any particular matrix norm, Bε(X) is the open ball with center at X and radius ε.
M∗n,d represents the open set of n × d complex matrices of full column rank and Grd(Cn) the grassmannian of subspaces of Cn of dimension d. In many works, Grd(Cn) is supposed to be provided with the topology induced by the “gap” metric (see, for example, [3]) that we will denote by Θ(·,·). It is well-known that Grd(Cn) can be identified with the homogeneous space M∗n,d/ Gld(C) (M∗n,d with the relative topology) through the action
M∗n,d × Gld(C) → M∗n,d (X,P) XP
With this identification, the gap topology has the same open sets as the quotient topology induced by the canonical projection
π : M ∗n , d → G r d ( C n ) X<X>
In addition, with these topologies, π is not only continuous but open. We will use these facts to approach the problem of the stability of A-invariant subspaces in a different manner to the one used in [3]. The following characterization of the invariant subspaces is important for our purpose.
Proposition 1.1. Let A ∈ Fn×n and let V ⊂ Fn be a subspace of dimension d. V is A-invariant if and only if there are matrices X ∈ Fn×d and L ∈ Fd×d such that rankX =d, V =<X > and AX =XL.
The proof is straightforward (see [2, p. 22]). Any matrix L satisfying the conditions of this Proposition will be called restriction matrix of A to V. Actually it is the restriction of A, as a linear map, to V with respect to any basis of Cn containing the columns of X as elements. This means, among other things, that once the matrix X has been fixed with its columns generating an A-invariant subspace, the restriction matrix L is unique (see again [2, p. 22]).
The above Proposition leads us to introduce the following set that will be very often used.
S(A) := {X ∈ M∗n,d|AX = XL for some matrix L ∈ Cd×d}. All this work is based on the following definition and theorem.