Rational Normal Form
May 11, 2007
Here is a brief overview of the main ideas of the statement and proof of
the theorem on rational normal form. Let
V
be a finite dimensional vector
space and let
T
be an operator on
V
.
A
T
invariant subspace
W
of
V
is said to be
cyclic
if there exists a
vector
w
∈
W
such that
W
is spanned by (
w, Tw, T
2
w,
· · ·
). If this is the
case, and if
m
is the smallest integer such that
T
m
w
belongs to the span of
(
w, Tw, T
2
w,
· · ·
, T
m

1
w
), then in fact
β
w
:= (
w, Tw, T
2
w,
· · ·
, T
m

1
w
) is a
basis for
W
. The matrix for
T
W
with respect to this basis is the companion
matrix of the characteristic polynomial
f
T
W
of
T
W
; furthermore in this case
f
T
W
is also the minimal polynomial of
T
W
.
Theorem:
If
T
is an operator on a finite dimensional vector space
V
,
then
V
can be written as a direct sum of cyclic subspaces
W
i
.
Moreover,
these subspaces can be chosen so that the characteristic polynomial of each
T
W
i
is a power of some irreducible factor of the minimal polynomial
p
T
of
T
.
Here is one way to attack the proof.
Let
p
T
:=
p
e
1
1
· · ·
p
e
r
r
be the factorization of
p
T
into irreducible factors.
Lemma:
Let
V
i
:= Ker
p
e
i
i
(
T
). Then
V
=
V
1
⊕
V
2
· · · ⊕
V
r
, and each
V
i
is nonzero.
Proof: Let
f
=
p
e
1
1
and let
g
=
i>
1
p
e
i
i
.
Since
f
and
g
have no common
factors, there are polynomials
p
and
q
such that
pf
+
qg
= 1. Now if
v
∈
V
, let
v
g
:=
p
(
T
)
f
(
T
)(
v
) and let
v
f
:=
q
(
T
)
g
(
T
)(
v
).
Then
v
=
v
f
+
v
g
.
Moreover,
f
(
T
)(
v
f
) =
f
(
T
)
g
(
T
)
q
(
T
)(
v
) = 0, so
v
f
∈
Ker
(
f
(
T
)). Similarly
v
g
∈
Ker
(
g
(
T
)).
Thus
V
= Ker(
f
(
T
)) + Ker(
g
(
T
)).
Moreover, if
v
∈
Ker(
f
(
T
))
∩
Ker(
g
(
T
)), then
v
=
p
(
T
)
f
(
T
)(
v
) +
q
(
T
)
g
(
T
)(
v
) = 0.
The
same equation shows that
q
(
T
) is inverse to
g
(
T
) on Ker(
f
(
T
)). Note that if
V
f
= 0, then
g
(
T
) = 0, which contradicts the minimality of
p
. Now proceed
by induction.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 GUREVITCH
 Linear Algebra, Vector Space, WI, direct sum, invariant subspace

Click to edit the document details