This preview shows pages 1–2. Sign up to view the full content.
Chapter CH.
The CayleyHamilton Theorem.
In this section
V
is a vector space over
F
=
R
or
C
of ﬁnite dimension
n
.
Deﬁne
M
0
n
=
M
0
n
(
F
) to be the set of matrices
A
= (
a
i,j
)
∈
M
n
such that
a
i,
1
= 0 for
all
i >
1. That is
A
is in
M
0
n
iﬀ all entries in the ﬁrst column of
A
, other than possibly
the ﬁrst entry, are 0.
Remark 1
. Let
A
= (
a
i,j
)
∈
M
n
. Then
A
∈
M
0
n
iﬀ
A
=
±
a
α
0
A
0
¶
where
a
=
a
(
A
)
∈
F
,
α
=
α
(
A
)
∈
M
1
,n

1
is a row matrix, and
A
0
∈
M
n

1
.
Lemma CH1.
Let
f
∈ L
(
V
)
,
X
=
{
x
1
,... ,x
n
}
a basis of
V
, and
A
=
m
X
(
f
)
. Then
A
∈
M
0
n
iﬀ
x
1
is an eigenvector for
f
.
Proof.
Let
A
= (
a
i,j
). As
A
=
m
X
(
f
),
x
1
is an eigenvector for
f
iﬀ
f
(
x
1
) =
a
1
,
1
x
1
iﬀ
a
1
,j
= 0 for
j >
1 iﬀ
A
∈
M
0
n
.
Lemma CH2.
Assume
A
∈
M
0
n
and let
a
=
a
(
A
)
. Then
char
A
(
x
) = (
x

a
)char
A
0
(
x
)
.
Proof.
Let
C
=
xI
n

A
= (
c
i,j
) and
C
0
=
xI
n

1

A
0
= (
c
0
i,j
). Then
C
=
±
x

a

α
0
xI
n

1

A
0
¶
=
±
x

a

α
0
C
0
¶
Hence as in Problem 3 on HW5,
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.
 Winter '07
 Aschbacher
 Linear Algebra, Algebra, Matrices, Vector Space

Click to edit the document details