Page 171 - Textos de Matemática Vol. 39

with ⎡⎢ y ⎤⎥
X′ = ⎢ yR ⎥ and
1n ⎣ . ⎦ yRn1−1
for some matrix R and some vector row y.
STABILITY OF INVARIANT SUBSPACES 161
In conclusion, X ∈ S(A) if and only if it can be written as ⎡⎢ X 1 ⎤⎥
X = ⎢X2⎥ ⎣ . ⎦
Xr
with Xi of the form (3.5) for some matrix L ∈ Cd×d and some row vector xi1. Assume now that there is a stable matrix X ∈ S(A). Then for any sequence of matrices {A′n} whose limit is A there is a sequence of matrices
Xn′ ∈ S(A′) whose limit is X.
Let A′n = A + ∆n where ∆n is a matrix whose entries are all zero except
the ones in positions (n1 + · · · + ni, n1 + · · · + ni + 1), i = 1, . . . , r, which are all equal to 1/n. It is clear that limn→∞ A′n = A. By using the same ideas as above, it is easily seen that X′ ∈ S(A′n) if and only if
⎡⎢ X 1′ n ⎤⎥ X′ =⎢X2′n⎥ n ⎣.⎦
X r′ n
⎡⎢ yRn1+···+ni−1 ⎤⎥
X′ = ni−1 ⎢yRn1+···+ni−1+1⎥,i = 2,...,r, in ⎣ . ⎦
yRn1+···+ni−1
In order {Xn′ } to be convergent it is necessary that Xi′n = 0 for
i=2,...,r.ThismeansthatyRp =0forp≥n1.Inthiscase,if lim Xn′ =X then n→∞
⎡⎢ X 1 ⎤⎥ ⎡⎢ y ⎤⎥ ⎢X2⎥ ⎢ yR ⎥
X =⎢⎣ . ⎥⎦ with X1 =⎢⎣ . ⎥⎦ and Xi =0, i=2,...,r. Xr yRn1−1
But, by Proposition 2.2 X is A-stable if and only if for any invertible matrix T , T X is T AT −1-stable. Since A has, at least, two Jordan blocks, there is a permutation matrix T, different from In, such that B = TAT−1 has the same Jordan blocks but with the first two blocks interchanged. For this matrix B we can construct a sequence Bn′ with entries 1/n in positions (n2, n2 + 1), (n2 +n1,n2 +n1 +1) and (n2 +n1 +···+ni,n2 +n1 +···+ni +1), i = 3,...,r. And the matrices Yn′ ∈ S(Bn′ ) would be like the matrices Xn′ but with the first