Section 9.3: Eigenvalues and Eigenvectors
Definition: Let A be an n × n matrix. A scalar λ is called an eigenvalue of A if there exists
a nonzero vector ~x such that
A~x = λ~x.
The vector ~x is called an eigenvector of A corresponding to λ.
The equation A~x = λ~x can be written in the form
(A − λI)~x = ~0.
Thus, λ is an eigenvalue of A if and only if this equation has a nontrivial solution. That is,
if and only if A − λI is singular, or equivalently,
det(A − λI) = 0.
If this determinant is expanded, we obtain an nth-degree polynomial of λ,
p(λ) = det(A − λI).
This polynomial is called the characteristic polynomial for the matrix A.
Example: Find the eigenvalues and eigenvectors of the matrix
1 2
A=
.
3 2
1
Example: Find the eigenvalues and eigenvectors of the matrix
3
6
A=
.
−1 −4
Determine the equations of the lines through the origin in the direction of the eigenvectors
~v1 and ~v2 , and graph the lines together with the eigenvectors and the vectors A~v1 and A~v2 .
2
Example: Find the eigenvalues and eigenvectors of the matrix
1 4
A=
.
1 −2
Example: Find the eigenvalues and eigenvectors of the matrix
1 1
A=
.
1 1
3
Example: Find the eigenvalues and eigenvectors of the matrix
2 3
A=
.
0 −1
Example: Find the eigenvalues and eigenvectors of the matrix
1 0
A=
.
4 3
Note: The eigenvalues of an upper- or lower-triangular matrix are the diagonal entries!
4
Example: Find the eigenvalues of the matrix
1 2
A=
.
−2 1
Definition: The trace of a 2 × 2 matrix
a b
A=
c d
is the scalar
tr(A) = a + d.
The trace and determinant of a 2 × 2 matrix are related to its eigenvalues.
Theorem: If A is a 2 × 2 matrix with eigenvalues λ1 and λ2 , then
tr A = λ1 + λ2
and
det A = λ1 λ2 .
Proof: The eigenvalues of A satisfy
a−λ
b
det(A − λI) = c
d−λ
= 0.
That is,
λ2 − (a + d)λ + (ad − bc) = 0
λ2 − (trA)λ + det A = 0.
If λ1 and λ2 are eigenvalues of A, then
(λ − λ1 )(λ − λ2 ) = 0
λ − (λ1 + λ2 )λ + λ1 λ2 = 0.
2
5
Theorem: (Routh-Hurwitz Criterion)
The real parts of the eigenvalues of a 2 × 2 matrix A are negative if and only if
tr A < 0
and
det A > 0.
Example: Consider the matrix
−2 3
A=
.
−1 1
Without explicitly computing the eigenvalues of A, determine whether the real parts of both
eigenvalues are negative.
Example: Consider the matrix
0 1
A=
.
2 −1
Without explicitly computing the eigenvalues of A, determine whether the real parts of both
eigenvalues are negative.
Note: The Routh-Hurwitz Criterion will be very useful in Chapter 11.
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Definition: Two nonzero vectors x~1 and x~2 are said to be linearly independent if there is
no scalar c such that x~1 = cx~2 . That is, x~1 and x~2 are not scalar multiples of each other.
Vectors which are not linearly independent are said to be linearly dependent.
Theorem: (Criterion for Linear Independence)
Let A be a 2 × 2 matrix with eigenvalues λ1 and λ2 , and respective eigenvectors v~1 and v~2 .
If λ1 6= λ2 , then v~1 and v~2 are linearly independent.
Note: A consequence of linear independence is that any vector can be written uniquely as a
linear combination of the two eigenvectors.
Suppose that v~1 and v~2 are linearly independent eigenvectors of a 2 × 2 matrix A. Then
any 2 × 1 vector ~x can be written as
~x = c1 v~1 + c2 v~2 ,
where c1 and c2 are uniquely determined constants. Applying A to the vector ~x yields
A~x = A(c1 v~1 + c2 v~2 )
= c1 Av~1 + c2 Av~2
= c1 λ1 v~1 + c2 λ2 v~2 .
This representation is particularly useful if we apply A repeatedly to ~x. In particular,
A2~x = A(c1 λ1 v~1 + c2 λ2 v~2 )
= c1 λ1 Av~1 + c2 λ2 Av~2
= c1 λ21 v~1 + c2 λ22 v~2 .
Continuing in this fashion, we have
An~x = c1 λn1 v~1 + c2 λn2 v~2 .
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Example: Consider the matrix
−2 1
A=
.
−4 3
(a) Show that ~v1 = h1, 1i and ~v2 = h1, 4i are linearly independent eigenvectors of A.
(b) Represent ~x = h−1, 2i as a linear combination of ~v1 and ~v2 .
(c) Use your results to compute A10~x.
8
Example: Consider the matrix
4 −3
A=
.
2 −1
If ~x = h−4, −2i, then compute A30~x without using a calculator.
9
The Leslie Matrix
Example: Suppose that a population is divided into two age classes with Leslie matrix
1.5 2
L=
.
0.08 0
Find the eigenvalues and eigenvectors of the Leslie matrix.
10
Theorem: (Eigenvalues and Eigenvectors of the Leslie Matrix)
Suppose that L is a 2 × 2 Leslie matrix with eigenvalues λ1 and λ2 .
1. The larger eigenvalue determines the growth parameter of the population.
2. The eigenvector corresponding to the larger eigenvalue is a stable age distribution.
Example: Suppose that a population is divided into two age classes with Leslie matrix
1 3
L=
.
0.7 0
(a) Find both eigenvalues.
(b) Give a biological interpretation of the larger eigenvalue.
(c) Find the stable age distribution.
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Example: Suppose that a population is divided into two age classes with Leslie matrix
0 5
L=
.
0.9 0
(a) Find both eigenvalues.
(b) Give a biological interpretation of the larger eigenvalue.
(c) Find the stable age distribution.
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