5.
1 Eigenvectors & Eigenvalues
Math 2331 – Linear Algebra
5.1 Eigenvectors & Eigenvalues
Jiwen He
Department of Mathematics, University of Houston
jiwenhe@math.uh.edu
math.uh.edu/∼jiwenhe/math2331
Jiwen He, University of Houston Math 2331, Linear Algebra 1 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
5.1 Eigenvectors & Eigenvalues
Eigenvectors & Eigenvalues
Eigenspace
Eigensvalues of Matrix Powers
Eigensvalues of Triangular Matrix
Eigenvectors and Linear Independence
Jiwen He, University of Houston Math 2331, Linear Algebra 2 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenvectors & Eigenvalues: Example
The basic concepts presented here - eigenvectors and eigenvalues -
are useful throughout pure and applied mathematics. Eigenvalues
are also used to study difference equations and continuous
dynamical systems. They provide critical information in
engineering design, and they arise naturally in such fields as
physics and chemistry.
Example
0 −2 1 −1
Let A = ,u= , and v = . Examine the
−4 2 1 1
images of u and v under multiplication by A.
Solution
0 −2 1 −2 1
Au = = = −2 = −2u
−4 2 1 −2 1
u is called an eigenvector of A since Au is a multiple of u.
Jiwen He, University of Houston Math 2331, Linear Algebra 3 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenvectors & Eigenvalues: Example (cont.)
0 −2 −1 −2
Av = = 6= λv
−4 2 1 6
v is not an eigenvector of A since Av is not a multiple of v.
Au = −2u, but Av 6= λv
Jiwen He, University of Houston Math 2331, Linear Algebra 4 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenvectors & Eigenvalues: Definition and Example
Eigenvectors & Eigenvalues
An eigenvector of an n × n matrix A is a nonzero vector x such
that Ax = λx for some scalar λ. A scalar λ is called an
eigenvalue of A if there is a nontrivial solution x of Ax = λx; such
an x is called an eigenvector corresponding to λ.
Example
0 −2
Show that 4 is an eigenvalue of A = and find the
−4 2
corresponding eigenvectors.
Solution: Scalar 4 is an eigenvalue of A if and only if Ax = 4x has
a nontrivial solution.
Ax−4x = 0
Ax−4 ( )x = 0
(A−4I ) x = 0.
Jiwen He, University of Houston Math 2331, Linear Algebra 5 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenvectors & Eigenvalues: Example (cont.)
To solve (A−4I ) x = 0, we need to find A−4I first:
0 −2 4 0 −4 −2
A−4I = − =
−4 2 0 4 −4 −2
Now solve (A−4I ) x = 0:
1 21 0
−4 −2 0
∼
−4 −2 0 0 0 0
1 1
− 2 x2 −2
⇒ x= = x2 .
x2 1
− 12
Each vector of the form x2 is an eigenvector corresponding
1
to the eigenvalue λ = 4.
Jiwen He, University of Houston Math 2331, Linear Algebra 6 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenvectors & Eigenvalues: Example (cont.)
Eigenspace for λ = 4
Warning
The method just used to find eigenvectors cannot be used to find
eigenvalues.
Eigenspace
The set of all solutions to (A−λI ) x = 0 is called the eigenspace
of A corresponding to λ.
Jiwen He, University of Houston Math 2331, Linear Algebra 7 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenspace: Example
Example
2 0 0
Let A = −1 3 1 . An eigenvalue of A is λ = 2. Find a
−1 1 3
basis for the corresponding eigenspace.
Solution:
2 0 0 0 0
A−2I = −1 3 1 −
0 0
−1 1 3 0 0
2− 0 0
= −1 3− 1
−1 1 3−
0 0
= −1 1
−1 1
Jiwen He, University of Houston Math 2331, Linear Algebra 8 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenspace: Example (cont.)
Augmented matrix for (A−2I ) x = 0:
0 0 0 0 1 −1 −1 0
−1 1 1 0 ∼ 0 0 0 0
−1 1 1 0 0 0 0 0
x1 x2 + x3 1 1
x = x2 = x2 = 1 + 0
x3 x3 0 1
So a basis for the eigenspace corresponding to λ = 2 is
1 1
1 , 0
0 1
Jiwen He, University of Houston Math 2331, Linear Algebra 9 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenspace: Example (cont.)
Effects of Multiplying Vectors in Eigenspaces for λ = 2 by A
Jiwen He, University of Houston Math 2331, Linear Algebra 10 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigensvalues of Matrix Powers: Example
Example
Suppose λ is eigenvalue of A. Determine an eigenvalue of A2 and
A3 . In general, what is an eigenvalue of An ?
Solution: Since λ is eigenvalue of A, there is a nonzero vector x
such that
Ax = λx.
Then
Ax = λx
A2 x = λAx
A2 x = λ x
A2 x = λ2 x
Therefore λ2 is an eigenvalue of A2 .
Jiwen He, University of Houston Math 2331, Linear Algebra 11 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigensvalues of Matrix Powers: Example (cont.)
Show that λ3 is an eigenvalue of A3 :
A2 x = λ2 x
A3 x = λ2 Ax
A3 x = λ3 x
Therefore λ3 is an eigenvalue of A3 .
In general, is an eigenvalue of An .
Jiwen He, University of Houston Math 2331, Linear Algebra 12 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigensvalues of Triangular Matrix
Theorem (1)
The eigenvalues of a triangular matrix are the diagonal entries.
Proof for the 3×3 Upper Triangular Case: Let
a11 a12 a13
A = 0 a22 a23 .
0 0 a33
a11 a12 a13 λ 0 0
A − λI = 0 a22 a23 − 0 λ 0
0 0 a33 0 0 λ
a11 − λ a12 a13
= 0 a22 − λ a23 .
0 0 a33 − λ
By definition, λ is an eigenvalue of A if and only if (A − λI ) x = 0
has a nontrivial solution. This occurs if and only if (A − λI ) x = 0
has a free variable. When does this occur?
Jiwen He, University of Houston Math 2331, Linear Algebra 13 / 14
5.1 Eigenvectors & Eigenvalues Definition Eigenspace Matrix Powers Triangular Matrix
Eigenvectors and Linear Independence
Theorem (2)
If v1 , . . . , vr are eigenvectors that correspond to distinct
eigenvalues λ1 , . . . , λr of an n × n matrix A, then {v1 , . . . , vr } is a
linearly independent set.
See the proof on page 307.
Jiwen He, University of Houston Math 2331, Linear Algebra 14 / 14
