Page 101 - Textos de Matemática Vol. 39

SPECTRALLY ARBITRARY FACTORIZATION: THE NONDEROGATORY CASE 91
Either A′ is 2-by-2, in which case it may be factored by Lemma 1.1; A′ is non-exceptional for some re-labelling of βm+1,...,βn,γm+1,...,γn and its fac- torizability is reduced to that case; or A′ may be further reduced in the manner that A has been so far. In any event, we may assume (inductively, if necessary) that A′ = B′C′, with B′ and C′ nonderogatory and having the desired eigen- values. Let
⎡11 ⎤ J = ⎢ ... ... ⎥
⎢ ⎢⎣ . . . 1 1 1 ⎥ ⎥⎦
be the m-by-m basic Jordan block associated with 1, and write I0=J−1 0J0
0A′ X B′ YC′ =I0.
Choosing Y = −B′−1XJ and X so that its last column is not in the column space of B′ − I (if that matrix is singular, and arbitrarily otherwise) ensures that the first factor is nonderogatory and has the desired eigenvalues. As no eigenvalue of C′ is 1, and C′ is nonderogatory, the second factor is nonderoga- tory, regardless of Y . But, because of the choice of Y , the two matrices factor the desired one.
The case m = n − 1 is similar by re-partitioning A as
writing
A=Im 0, 0...0 t λ
J−1 0J 0 A=xβn yγn,
XJ+B′Y A′
choosingy= 1 ([00...t]−xJ)andx=[00...01]. βn
If m = n, λ must be 1 and we write (for t ̸= 1, which we may assume) ⎛⎡0 ··· 0⎤⎞
⎜ ⎢ . ··· . ⎥⎟
A = ⎜⎝ J + ⎢⎣ 0 · · · 0 ⎥⎦ ⎟⎠ J − 1
0··· 0 tt0
to give the desired factorization, which complete the proof.