Page 172 - Textos de Matemática Vol. 39

162 ION ZABALLA
block of size n2 × d and the second one of size n1 × d. As TX is B-stable, we
w o u l d h a v e ⎡⎢ Y 1 ⎤⎥ ⎢Y2⎥
⎡⎢ z ⎤⎥ ⎢ zR ⎥
Yi =0,i=2,...,r.
TX =Y′ =⎢⎣ . ⎥⎦ Yr
Y1 =⎢⎣ . ⎥⎦ and zRn2−1
with
But T permutes the first and second blocks. Thus
which is a contradiction.
⎣ . ⎦ 0

⎡⎢ ⎢ X0 ⎤⎥ ⎥ TX=⎢ .1⎥,
Bearing in mind the remarks at the end of Section 2 we have the following theorem characterizing the stable invariant subspaces of any endomorphism. Recall that a subspace is called radical if it is a direct sum of root subspaces.
Theorem 3.3. Let A ∈ Cn×n. If A is nonderogatory then any A-invariant subspace is stable; otherwise, the only A-invariant subspaces which are stable are {0} and the radical subspaces.
References
[1] F. R. Gantmacher, The theory of matrices, Vol. I, Chelsea Publishing Company, New York, 1959.
[2] G. W. Stewart, Ji-guang Sun, Matrix perturbation theory, Academic Press, New York, 1990.
[3] I. Gohberg. P. Lancaster, L. Rodman, Invariant subspaces of matrices and applications John Wiley and Sons. New York, 1986.
Departamento de Matema´tica Aplicada y EIO
Universidad del Pa´ıs Vasco-Euskal Herriko Unibertsitatea Apdo. Correos 644, Bilbao 48080,Spain
E-mail address: ion.zaballa@ehu.es