Page 210 - Advanced Linear Algebra

194 Advanced Linear Algebra

Now suppose that -~ s and ²%³ ~ % b % b ! is an irreducible quadratic.

8
If is a -cyclic basis for > , then
c !
´µ ~ > 8 ?
c
However, there is a more appealing matrix representation of . To this end, let

( ( be the matrix above. As a complex matrix, has two distinct eigenvalues:
j ! c
~c f

Now, a matrix of the form

c
)~ > ?

has characteristic polynomial ²%³ ~ ²% c ³ b and eigenvalues f . So
if we set
j ! c
~c and ~c

then has the same two distinct eigenvalues as and so and have the
)
)
(
(
same Jordan canonical form over . It follows that and are similar over d
d
(
)
and therefore also over , by Theorem 7.20. Thus, there is an ordered basis 9
s
for which ´µ ~ ) .
9
Theorem 8.8 If -~ s and > is cyclic and deg ² ²%³³ ~ , then there is an

ordered basis for which
9
c
´µ ~ > 9 ?

Now we can proceed with the real version of Schur's theorem. For the sake of
the exposition, we make the following definition.
Definition A matrix ( 4 ²-³ is almost upper triangular if it has the form

v ( i y
x ( {
(~ x {
Æ
w ( z block

where