Page 211 - Advanced Linear Algebra

Eigenvalues and Eigenvectors 195

c
(~ ´ µ or (~ > ?

for Á  - . A linear operator  ²= ³ is almost upper triangularizable if
B
8
there is an ordered basis for which ´µ 8 is almost upper triangular.
To see that every real linear operator is almost upper triangularizable, we use
Theorem 7.19, which states that if ²%³ is a prime factor of ²%³ , then = has a

cyclic submodule > of order ² % ³ . Hence, > is a -cyclic subspace of

dimension deg² ²%³³ and O > has characteristic polynomial ²%³ .
Now, the minimal polynomial of a real operator ²= ³ factors into a product

B
-
of linear and irreducible quadratic factors. If ²%³ has a linear factor over ,

then = has a one-dimensional -invariant subspace > . If ² % ³ has an
irreducible quadratic factor ²%³ , then = has a cyclic submodule > of order
²%³ and so a matrix representation of on > is given by the matrix

c
(~ > ?

This is the basis for an inductive proof, as in the complex case.
Theorem 8.9 (Schur's theorem: real case ) If is a real vector space, then
=
every linear operator on is almost upper triangularizable.
=
Proof. As with the complex case, it is simpler to proceed using matrices, by
(
showing that any d real matrix is similar to an almost upper triangular
matrix. The result is clear if ~ . Assume for the purposes of induction that
any square matrix of size less than d is almost upper triangularizable.

We have just seen that - has a one-dimensional ( -invariant subspace > or a
-cyclic subspace > has irreducible characteristic
two-dimensional ( , where (
polynomial on > . Hence, we may choose a basis for - for which the first
8
one or first two vectors are a basis for > . Then
( i
´ µ ( 8 ~ > ?
( block
where

c
(~ ´ µ or (~ > ?

and ( has size d . The induction hypothesis applied to ( gives an
for which
invertible matrix 7 4
<~ 7( 7 c