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)
2.
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.
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.
y
x
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
e1
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 Ae1
6.
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
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
e2
x
e1 Ae1
Ae2
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
v
x
e1 Ae1
Ae2
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
v
x
e1 Ae1
Ae2
10.
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
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
Av
12.
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
13.
Example
Draw the eﬀect of the linear transformation which is multiplication
by A2 .
y
x
14.
Example
Draw the eﬀect of the linear transformation which is multiplication
by A2 .
y
x
15.
Example
Draw the eﬀect of the linear transformation which is multiplication
by A2 .
y
x
16.
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
17.
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
18.
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
19.
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
20.
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
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
e2 Be1
v
Be2
x
e1
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
e2 Be1
v
Be2
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
e2 Be1
v
Be2
x
e1
Bv
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
v
Be2
x
e1
Bv
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
v
Be2
x
e1
Bv
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
x
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
v1
x
28.
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
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
Bv1
v1
x
v2
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
Bv1
Bv2
v1
x
v2
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
Bv1
Bv2
v1
x
2e2
v2
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
Bv1
Bv2
v1
x
2e2
v2
33.
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
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
2Be2
Bv1
Bv2
v1
x
2e2
v2
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
2Be2
Bv1
Bv2
v1
x
2e2
v2
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
2Be2
Bv1
Bv2
v1
x
2e2
v2
37.
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
38.
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.
39.
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.
40.
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.
41.
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
42.
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
43.
Example (Worksheet Problem 2)
The number −2 is also an eigenvalue for A. Find an eigenvector.
44.
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
45.
Summary
To ﬁnd the eigenvector(s) of a matrix corresponding to an
eigenvalue λ, do Gaussian Elimination on A − λI.
46.
Find the eigenvalues of a matrix
Example (Worksheet Problem 3)
Determine the eigenvalues of
−4 −3
A= .
3 6
47.
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.
48.
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?
49.
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!
50.
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.
51.
−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.
52.
We can ﬁnd the eigenvectors now, based on what we did before.
We have
A + 3I =
53.
We can ﬁnd the eigenvectors now, based on what we did before.
We have
−1 −3
A + 3I =
3 9
54.
We can ﬁnd the eigenvectors now, based on what we did before.
We have
−1 −3 13
A + 3I =
3 9 00
55.
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
56.
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 =
57.
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
58.
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
59.
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
60.
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.
61.
Example (Worksheet Problem 4)
Find the eigenvalues of
−10 6
A= .
−15 9
62.
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
63.
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.
64.
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
65.
Applications
In a Markov Chain, the steady-state vector is an eigenvector
corresponding to the eigenvalue 1.
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