MA 3280 Lecture 23  More on Eigenvectors
Wednesday, April 2, 2014.
Objectives: Discuss the diagonalization of a matrix.
There is something called a diagonalization for a matrix. This comes easily from the eigenvalues and
eigenvectors. For the matrix
(1)
A
=
b
1

1
2
4
B
.
We got the eigenvalues
λ
= 2
,
3, and eigenvectors of the form
(2)
b

a
a
B
and
b

a
2
a
B
.
We can diagonalize
A
by putting any particular eigenvectors for each eigenvalue, for example
(3)
b
1

1
B
and
b
1

2
B
,
as columns in a matrix
(4)
P
=
b
1
1

1

2
B
,
±nding the inverse
(5)
P

1
=
1

2

(

1)
b

2

1
1
1
B
=
b
2
1

1

1
B
.
We get
D
=
P

1
AP
, or
(6)
b
2
1

1

1
B b
1

1
2
4
B b
1
1

1

2
B
=
b
4
2

3

3
B b
1
1

1

2
B
=
b
2
0
0
3
B
.
Linear maps
We can de±ne a function from an
R
m
to an
R
n
by multiplying vectors by a matrix. For example, we can
de±ne the function
L
:
R
2
→
R
2
by
(7)
L
pb
x
1
x
2
B P
=
b
2
0
0
3
B b
x
1
x
2
B
=
b
2
x
1
3
x
2
B
,
which takes vectors in
R
2
and maps them to vectors in
R
2
. This function is an example of something we’ll
call a
linear map
.
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