Page 209 - Advanced Linear Algebra

Eigenvalues and Eigenvectors 193

Proof. Part 2) follows from part 1). The proof of part 1) is most easily
accomplished by matrix means, namely, we prove that every square matrix
( 4 ²-³ whose characteristic polynomial splits over is similar to an upper
-

triangular matrix. If ~ there is nothing to prove, since all d matrices are
upper triangular. Assume the result is true for c and let (  4 ²-³ .

#
(
Let be an eigenvector associated with the eigenvalue  - of and extend
¸# ¹ to an ordered basis 8 ~ ²# ÁÃÁ# ³ for s . The matrix of ( with respect

to has the form
8
i

´ µ ( 8 ~ > ?
( block
(
for some ( 4 c ²-³ . Since ´ ( µ 8 and are similar, we have
det ²%0 c (³ ~ det ²%0 c ´ ( 8 ³ det ²%0 c ( ³
µ ³ ~ ²% c

Hence, the characteristic polynomial of ( also splits over and the induction
-
²-³ for which
hypothesis implies that there is an invertible matrix 7 4 c
<~ 7( 7 c

is upper triangular. Hence, if

8~ > ?
7 block
then is invertible and
8
8´(µ 8 c ~ > ? > i ? > c ? ~ > i ?
8
7 ( 7 <
is upper triangular.

The Real Case

When the base field is -~ s , an operator is upper triangularizable if and
only if its characteristic polynomial splits over . (Why?) We can, however,
s
always achieve a form that is close to triangular by permitting values on the first
subdiagonal.

Before proceeding, let us recall Theorem 7.11, which says that for a module >
of prime order ²%³ , the following are equivalent:

) is cyclic
1 >
) is indecomposable
2 >
)
3 ²%³ is irreducible

)
4 is nonderogatory, that is, ²%³ ~ ²%³

5)dim²> ³ ~ deg² ²%³³ .