1. Lesson 14
Eigenvalues and Eigenvectors
Math 20
October 22, 2007
Announcements
Midterm almost done
Problem Set 5 is on the WS. Due October 24
OH: Mondays 1–2, Tuesdays 3–4, Wednesdays 1–3 (SC 323)
Prob. Sess.: Sundays 6–7 (SC B-10), Tuesdays 1–2 (SC 116)

3. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.

4. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
x

5. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
x
e1

7. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
x
e1 Ae1

8. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
e2
x
e1 Ae1

9. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
e2
x
e1 Ae1
Ae2

11. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
e2
v
x
e1 Ae1
Ae2

14. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
e2
v
x
e1 Ae1
Ae2

15. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
e2
v
x
e1 Ae1
Ae2
Av

16. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
e2
v
x
e1 Ae1
Ae2
Av

17. Geometric effect of a diagonal linear transformation
Example
20
Let A = . Draw the effect of the linear transformation
0 −1
which is multiplication by A.
y
e2
v
x
e1 Ae1
Ae2
Av

18. Example
Draw the effect of the linear transformation which is multiplication
by A2 .
y
x

19. Example
Draw the effect of the linear transformation which is multiplication
by A2 .
y
x

20. Example
Draw the effect of the linear transformation which is multiplication
by A2 .
y
x

21. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
x

22. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
x
e1

23. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
Be1
x
e1

24. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
e2 Be1
x
e1

25. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
e2 Be1
Be2
x
e1

26. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
e2 Be1
v
Be2
x
e1

27. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
e2 Be1
v
Be2
x
e1

29. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
e2 Be1
v
Be2
x
e1
Bv

30. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
e2 Be1
v
Be2
x
e1
Bv

31. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
e2 Be1
v
Be2
x
e1
Bv

32. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
x

34. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
v1
x

35. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
Bv1
v1
x

36. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
Bv1
v1
x
v2

38. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
Bv1
Bv2
v1
x
v2

39. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
Bv1
Bv2
v1
x
2e2
v2

40. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
Bv1
Bv2
v1
x
2e2
v2

41. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
Bv1
Bv2
v1
x
2e2
v2

42. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
2Be2
Bv1
Bv2
v1
x
2e2
v2

43. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
2Be2
Bv1
Bv2
v1
x
2e2
v2

44. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
2Be2
Bv1
Bv2
v1
x
2e2
v2

45. Geometric effect of a non-diagonal linear transformation
Example
1/2 3/2
Let B = . Draw the effect of the linear transformation
3/2 1/2
which is multiplication by B.
y
2Be2
Bv1
Bv2
v1
x
2e2
v2

46. The big concept
Definition
Let A be an n × n matrix. The number λ is called an eigenvalue
of A if there exists a nonzero vector x ∈ Rn such that
Ax = λx. (1)
Every nonzero vector satisfying (1) is called an eigenvector of A
associated with the eigenvalue λ.

48. Finding the eigenvector(s) corresponding to an eigenvalue
Example (Worksheet Problem 1)
0 −2
Let A = . The number 3 is an eigenvalue for A. Find
−3 1
an eigenvector corresponding to this eigenvalue.

50. Finding the eigenvector(s) corresponding to an eigenvalue
Example (Worksheet Problem 1)
0 −2
Let A = . The number 3 is an eigenvalue for A. Find
−3 1
an eigenvector corresponding to this eigenvalue.
Solution
We seek x such that
Ax = 3x =⇒ (A − 3I)x = 0.

51. Finding the eigenvector(s) corresponding to an eigenvalue
Example (Worksheet Problem 1)
0 −2
Let A = . The number 3 is an eigenvalue for A. Find
−3 1
an eigenvector corresponding to this eigenvalue.
Solution
We seek x such that
Ax = 3x =⇒ (A − 3I)x = 0.
−3 −2 0 1 2/3 0
A − 3I 0 =
−3 −2 0 0 00

53. Finding the eigenvector(s) corresponding to an eigenvalue
Example (Worksheet Problem 1)
0 −2
Let A = . The number 3 is an eigenvalue for A. Find
−3 1
an eigenvector corresponding to this eigenvalue.
Solution
We seek x such that
Ax = 3x =⇒ (A − 3I)x = 0.
−3 −2 0 1 2/3 0
A − 3I 0 =
−3 −2 0 0 00
−2/3 −2
So , or , are possible eigenvectors.
1 3

55. Example (Worksheet Problem 2)
The number −2 is also an eigenvalue for A. Find an eigenvector.

56. Example (Worksheet Problem 2)
The number −2 is also an eigenvalue for A. Find an eigenvector.
Solution
We have
2 −2 1 −1
A + 2I = .
−3 3 00
1
is an eigenvector for the eigenvalue −2.
So
1

57. Summary
To find the eigenvector(s) of a matrix corresponding to an
eigenvalue λ, do Gaussian Elimination on A − λI.

58. Find the eigenvalues of a matrix
Example (Worksheet Problem 3)
Determine the eigenvalues of
−4 −3
A= .
3 6

59. Find the eigenvalues of a matrix
Example (Worksheet Problem 3)
Determine the eigenvalues of
−4 −3
A= .
3 6
Solution
Ax = λx has a nonzero solution if and only if (A − λI)x = 0 has a
nonzero solution, if and only if A − λI is not invertible.

62. Find the eigenvalues of a matrix
Example (Worksheet Problem 3)
Determine the eigenvalues of
−4 −3
A= .
3 6
Solution
Ax = λx has a nonzero solution if and only if (A − λI)x = 0 has a
nonzero solution, if and only if A − λI is not invertible. What
magic number determines whether a matrix is invertible?

63. Find the eigenvalues of a matrix
Example (Worksheet Problem 3)
Determine the eigenvalues of
−4 −3
A= .
3 6
Solution
Ax = λx has a nonzero solution if and only if (A − λI)x = 0 has a
nonzero solution, if and only if A − λI is not invertible. What
magic number determines whether a matrix is invertible? The
determinant!

64. Find the eigenvalues of a matrix
Example (Worksheet Problem 3)
Determine the eigenvalues of
−4 −3
A= .
3 6
Solution
Ax = λx has a nonzero solution if and only if (A − λI)x = 0 has a
nonzero solution, if and only if A − λI is not invertible. What
magic number determines whether a matrix is invertible? The
determinant! So to find the possible values λ for which this could
be true, we need to find the determinant of A − λI.

65. −4 − λ −3
det(A − λI) =
6−λ
3
= (−4 − λ)(6 − λ) − (−3)(3)
= (−24 − 2λ + λ2 ) + 9 = λ2 − 2λ − 15
= (λ + 3)(λ − 5)
So λ = −3 and λ = 5 are the eigenvalues for A.

66. We can find the eigenvectors now, based on what we did before.
We have
A + 3I =

67. We can find the eigenvectors now, based on what we did before.
We have
−1 −3
A + 3I =
3 9

68. We can find the eigenvectors now, based on what we did before.
We have
−1 −3 13
A + 3I =
3 9 00

69. We can find the eigenvectors now, based on what we did before.
We have
−1 −3 13
A + 3I =
3 9 00
−3
So is an eigenvector for this eigenvalue.
1

70. We can find the eigenvectors now, based on what we did before.
We have
−1 −3 13
A + 3I =
3 9 00
−3
So is an eigenvector for this eigenvalue. Also,
1
A − 5I =

71. We can find the eigenvectors now, based on what we did before.
We have
−1 −3 13
A + 3I =
3 9 00
−3
So is an eigenvector for this eigenvalue. Also,
1
−9 −3
A − 5I =
3 1

72. We can find the eigenvectors now, based on what we did before.
We have
−1 −3 13
A + 3I =
3 9 00
−3
So is an eigenvector for this eigenvalue. Also,
1
−9 −3 1 1/3
A − 5I =
3 1 0 0

73. We can find the eigenvectors now, based on what we did before.
We have
−1 −3 13
A + 3I =
3 9 00
−3
So is an eigenvector for this eigenvalue. Also,
1
−9 −3 1 1/3
A − 5I =
3 1 0 0
−1/3 −1
So or would be an eigenvector for this eigenvalue.
1 3

74. Summary
To find the eigenvalues of a matrix A, find the determinant of
A − λI. This will be a polynomial in λ, and its roots are the
eigenvalues.

75. Example (Worksheet Problem 4)
Find the eigenvalues of
−10 6
A= .
−15 9

76. Example (Worksheet Problem 4)
Find the eigenvalues of
−10 6
A= .
−15 9
Solution
The characteristic polynomial is
−10 − λ 6
= (−10−λ)(9−λ)−(−15)(6) = λ2 +λ = λ(λ+1).
−15 9−λ

77. Example (Worksheet Problem 4)
Find the eigenvalues of
−10 6
A= .
−15 9
Solution
The characteristic polynomial is
−10 − λ 6
= (−10−λ)(9−λ)−(−15)(6) = λ2 +λ = λ(λ+1).
−15 9−λ
The eigenvalues are 0 and −1.

78. Example (Worksheet Problem 4)
Find the eigenvalues of
−10 6
A= .
−15 9
Solution
The characteristic polynomial is
−10 − λ 6
= (−10−λ)(9−λ)−(−15)(6) = λ2 +λ = λ(λ+1).
−15 9−λ
3
The eigenvalues are 0 and −1. A set of eigenvectors are and
5
2
.
3

79. Applications
In a Markov Chain, the steady-state vector is an eigenvector
corresponding to the eigenvalue 1.