ch6 - CHAPTER 6 EIGENVALUES AND EIGENVECTORS SECTION 6.1...

Section 6.1 289
CHAPTER 6
EIGENVALUES AND EIGENVECTORS
SECTION 6.1
INTRODUCTION TO EIGENVALUES
In each of Problems 1–32 we first list the characteristic polynomial ( ) p λ λ = − A I of the given
matrix A , and then the roots of ( ) p λ — which are the eigenvalues of A . All of the eigenvalues
that appear in Problems 1–26 are integers, so each characteristic polynomial factors readily. For
each eigenvalue
j
λ of the matrix A , we determine the associated eigenvector(s) by finding a basis
for the solution space of the linear system
( )
.
j
λ − = A I v 0 We write this linear system in scalar
form in terms of the components of
[ ]
.
T
a b = v In most cases an associated eigenvector is
then apparent. If A is a 2 2 × matrix, for instance, then our two scalar equations will be multiples
one of the other, so we can substitute a convenient numerical value for the first component a of v
and then solve either equation for the second component b (or vice versa).
1. Characteristic polynomial:
2
( ) 5 6 ( 2)( 3) p λ λ λ λ λ = − + = − −
Eigenvalues:
1 2
2, 3 λ λ = =
With
1
2: λ =
2 2 0
0
a b
a b
− =


− =

1
1
1
 
=
 
 
v
With
2
3: λ =
2 0
2 0
a b
a b
− =


− =

2
2
1
 
=
 
 
v
2. Characteristic polynomial: 2 ( ) 2 ( 1)( 2) p λ λ λ λ λ = − − = + −
Eigenvalues:
1 2
1, 2 λ λ = − =
With
1
1: λ = −
6 6 0
3 3 0
a b
a b
− =


− =

1
1
1
 
=
 
 
v
With
2
2: λ =
3 6 0
3 6 0
a b
a b
− =


− =

2
2
1
 
=
 
 
v

290 Chapter 6
3. Characteristic polynomial: 2 ( ) 7 10 ( 2)( 5) p λ λ λ λ λ = − + = − −
Eigenvalues:
1 2
2, 5 λ λ = =
With
1
2: λ =
6 6 0
3 3 0
a b
a b
− =


− =

1
1
1
 
=
 
 
v
With
2
5: λ =
3 6 0
3 6 0
a b
a b
− =


− =

2
2
1
 
=
 
 
v
4. Characteristic polynomial: 2 ( ) 3 2 ( 1)( 2) p λ λ λ λ λ = − + = − −
Eigenvalues:
1 2
2, 5 λ λ = =
With
1
1: λ =
3 3 0
2 2 0
a b
a b
− =


− =

1
1
1
 
=
 
 
v
With
2
2: λ =
2 3 0
2 3 0
a b
a b
− =


− =

2
3
2
 
=
 
 
v
5. Characteristic polynomial: 2 ( ) 5 4 ( 1)( 4) p λ λ λ λ λ = − + = − −
Eigenvalues:
1 2
1, 4 λ λ = =
With
1
1: λ =
9 9 0
6 6 0
a b
a b
− =


− =

1
1
1
 
=
 
 
v
With
2
4: λ =
6 9 0
6 9 0
a b
a b
− =


− =

2
3
2
 
=
 
 
v
6. Characteristic polynomial:
2
( ) 5 6 ( 2)( 3) p λ λ λ λ λ = − + = − −
Eigenvalues:
1 2
1, 4 λ λ = =
With
1
2: λ =
4 4 0
3 3 0
a b
a b
− =


− =

1
1
1
 
=
 
 
v
With
2
3: λ =
3 4 0
3 4 0
a b
a b
− =


− =

2
4
3
 
=
 
 
v
7. Characteristic polynomial: 2 ( ) 6 8 ( 2)( 4) p λ λ λ λ λ = − + = − −
Eigenvalues:
1 2
2, 4 λ λ = =

Section 6.1 291
With
1
2: λ =
8 8 0
6 6 0
a b
a b
− =


− =

1
1
1
 
=
 
 
v
With
2
4: λ =
6 8 0
6 8 0
a b
a b
− = 

− =

2
4
3
 
=
 
 
v
8. Characteristic polynomial: 2 ( ) 3 2 ( 2)( 1) p λ λ λ λ λ = + + = + +
Eigenvalues:
1 2
2, 1 λ λ = − = −
With
1
2: λ = −
9 6 0
12 8 0
a b
a b
− =


− =

1
2
3
 
=
 
 
v
With
2
1: λ = −
8 6 0
12 9 0
a b
a b
− =


− =

2
3
4
 
=
 
 
v
9.