Previous lecture
Definition
V
, finite-dimensional vector space
T
:
V
→
V
, linear
Then
1
T
is called
diagonalizable
if there is an ordered basis
β
for
V
such that
[
T
]
β
is a diagonal matrix.
2
A
square
matrix
A
is called
diagonalizable
if the left-multiplication
L
A
is diagonalizable.
()
MH1201
3 / 3

Remark
MH1201
10 / 68

Remark
Suppose that
D
= [
T
]
β
is a diagonal matrix.
D
= [
T
]
β
=
D
11
. . .
0
. . .
0
. . .
. . .
. . .
. . .
. . .
0
. . .
D
jj
. . .
0
. . .
. . .
. . .
. . .
. . .
0
. . .
0
. . .
D
nn
.
MH1201
10 / 68

Remark
Suppose that
D
= [
T
]
β
is a diagonal matrix.
D
= [
T
]
β
=
D
11
. . .
0
. . .
0
. . .
. . .
. . .
. . .
. . .
0
. . .
D
jj
. . .
0
. . .
. . .
. . .
. . .
. . .
0
. . .
0
. . .
D
nn
.
Then for each vector
v
j
∈
β
, taking into account that the
j
th
column of
D
is just
[
T
(
v
j
)]
β
, we have
T
(
v
j
) =
D
1
j
v
1
+
D
2
j
v
2
+
· · ·
+
D
jj
v
j
+
· · ·
+
D
nj
v
n
=
D
jj
v
j
=
λ
j
v
j
,
where
λ
j
=
D
jj
.
MH1201
10 / 68

MH1201
11 / 68

These observations motivate the following definitions.
MH1201
12 / 68

These observations motivate the following definitions.
Definition
MH1201
12 / 68

These observations motivate the following definitions.
Definition
1
Let
T
be in
L
(
V
)
.
MH1201
12 / 68

These observations motivate the following definitions.
Definition
1
Let
T
be in
L
(
V
)
.
A
non-zero
vector
v
∈
V
is called an
eigenvector
of
T
if there
exists a scalar
λ
such that
T
(
v
) =
λ
v
.
MH1201
12 / 68

These observations motivate the following definitions.
Definition
1
Let
T
be in
L
(
V
)
.
A
non-zero
vector
v
∈
V
is called an
eigenvector
of
T
if there
exists a scalar
λ
such that
T
(
v
) =
λ
v
.
The scalar
λ
is called the
eigenvalue
corresponding to the
eigenvector
v
.
MH1201
12 / 68

These observations motivate the following definitions.
Definition
1
Let
T
be in
L
(
V
)
.
A
non-zero
vector
v
∈
V
is called an
eigenvector
of
T
if there
exists a scalar
λ
such that
T
(
v
) =
λ
v
.
The scalar
λ
is called the
eigenvalue
corresponding to the
eigenvector
v
.
2
Let
A
be in
M
n
×
n
(
R
)
.
MH1201
12 / 68

These observations motivate the following definitions.
Definition
1
Let
T
be in
L
(
V
)
.
A
non-zero
vector
v
∈
V
is called an
eigenvector
of
T
if there
exists a scalar
λ
such that
T
(
v
) =
λ
v
.
The scalar
λ
is called the
eigenvalue
corresponding to the
eigenvector
v
.
2
Let
A
be in
M
n
×
n
(
R
)
.
A
non-zero
vector
v
∈
R
n
is called an
eigenvector
of
A
if
v
is an
eigenvector of
L
A
; that is, if
Av
=
λ
v
for some scalar
λ
.
MH1201
12 / 68

These observations motivate the following definitions.
Definition
1
Let
T
be in
L
(
V
)
.
A
non-zero
vector
v
∈
V
is called an
eigenvector
of
T
if there
exists a scalar
λ
such that
T
(
v
) =
λ
v
.
The scalar
λ
is called the
eigenvalue
corresponding to the
eigenvector
v
.
2
Let
A
be in
M
n
×
n
(
R
)
.
A
non-zero
vector
v
∈
R
n
is called an
eigenvector
of
A
if
v
is an
eigenvector of
L
A
; that is, if
Av
=
λ
v
for some scalar
λ
.
The scalar
λ
is called the
eigenvalue
of
A
corresponding to the
eigenvector
v
.
MH1201
12 / 68

Note that a vector is an eigenvector of a matrix
A
if and only if
it is an
eigenvector of
L
A
.
MH1201
13 / 68

Note that a vector is an eigenvector of a matrix
A
if and only if
it is an