Jordan Decomposition Theorem
: LinearAlg
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
squash@ufl.edu
Webpage
http://www.math.uﬂ.edu/
∼
squash/
7 November, 2010 (at
18:32
)
Abstract:
Gives a homegrown proof of the Jor
dan Decomposition Theorem. (
Some of the lemmas work
in Hilbert space.
)
The “Partialform JCF Theorem”,
(26), needs to be reworked.
Prolegomenon
Our goal is to prove the
“
JCF
”
(
Jordan Canonical
Form
) Theorem
for a linear trn
T
:
H
→
H
, where
H
is a ﬁnitedim’al vectorspace. Formally, we’ll
assume that
H
is
F
×H
, where the ﬁeld
F
is ei
ther
R
or
C
.
For vectorspaces use
vectorspace:
H
,
A
,
B
,
E
,
V
dimension:
H
,
A
,
B
,
E
,
V
.
Use sansserif font for matrices
A
,
B
,
G
,
I
,
M
. For
square matrices
A
e
, let
Diag
(
A
1
,...,
A
E
) be the
partitioned matrix which has
A
1
A
E
along its
diagonal, and zeros elsewhere.
1
: Notation
.
A collection
C
:=
{
V
1
,
V
2
V
L
}
of
subspaces
of
H
is
linearly independent
(
abbreviation
linindep
) if the only soln to
v
1
+
···
+
v
L
=
0
, with each
v
‘
∈
V
‘
,
is the trivial soln
v
1
=
0
v
L
=
0
.
Recall that a subspace
V
⊂
H
is
T
invariant
if
T
(
V
)
⊂
V
.
I’ll use
eval
,
evec
and
espace
for
eigenvalue
,
eigenvector
and
eigenspace
.
±
§
1 Examining nilpotent case
In sections
§
1 and
§
2, “eval” means the eigen
value
zero
, and “evec” means an eigenvector with
eval
zero
.
2
: Defn
.
W.r.t
T
, a vector
v
is
nilpotent
if
T
d
(
v
) =
0
for some posint
d
. Indeed, the
T

depth
of a vector
v
, written
T
Depth(
v
), is the
inﬁmum of all natnums
n
for which
T
n
(
v
) is
0
.
The zerovector has depth 0. An evec for eval=0
has depth 1. (
A nonnilpotent vector has depth
∞
.
)
Use Nil(
T
) for the
nilspace
of
T
; it comprises
the set of ﬁnitedepth vectors. So
Nil(
T
)
def
==
∞
[
n
=1
Ker(
T
n
)
note
⊃
Ker(
T
)
.
Transformation
T
is
nilpotent
if there exists a
posint
D
such that
T
D
=
0
. Since
H
is ﬁnite di
mensional, trn
±
²
³
´
T
is nilpotent
iﬀ
Nil(
T
) =
H
.
±
3
:
Depth Lemma (
preliminary
).
Consider a sum
v
1
+
v
2
+
+
v
L
3
0
:
whose depths satisfy
d
1
> d
2
> ... > d
L
.
Then the depth of
(3
0
)
is
d
1
.
Proof.
Exercise.
♦
A
downtup
(
“down tuple”
)
→
D
= (
D
1
,...,D
E
)
is a sequence of integers with
D
1
>
D
2
>
...
>
D
E
>
1
.
4
:
The
size
of
D
is the sum
D
1
+
+
D
E
.
A
posint
D
determines a
D
×
D
Jordan Block
ma
trix
JB
(
D
) :=
0 1
0 1
0 1
.
.
.
.
.
.
0 1
0
5
:
with zeros on the diagonal and ones on the ﬁrst
oﬀdiagonal. Every undisplayed position is zero.
6
:
Nilpotent JCF Theorem.
A nilpotent
T
:
F
²
has a
unique
downtup
D
so that
M
=
M
(
D
) :=
Diag
±
(
D
1
)
(
D
E
)
²
7
:
is the matrix of
T
w.r.t some
basis. In particular,
Size(
D
)
equals
H
.
♦
Webpage
http://www.math.uﬂ.edu/
∼
squash/
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