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Section 5.3 Eigenvalues and Characteristic Polynomials (GT3)

Subsection 5.3.1 Warm Up

Activity 5.3.1.

Let R:R2→R2 be the transformation given by rotating vectors about the origin through and angle of 45∘, and let S:R2→R2 denote the transformation that reflects vectors about the line x1=x2.
(a)
If L is a line, let R(L) denote the line obtained by applying R to it. Are there any lines L for which R(L) is parallel to L?
(b)
Now consider the transformation S. Are there any lines L for which S(L) is parallel to L?

Subsection 5.3.2 Class Activities

Activity 5.3.2.

An invertible matrix M and its inverse M−1 are given below:
M=[1234]M−1=[−213/2−1/2]
Which of the following is equal to det(M)det(M−1)?
  1. −1
  2. 0
  3. 1
  4. 4

Observation 5.3.4.

Consider the linear transformation A:R2→R2 given by the matrix A=[2203].
Figure 59. Transformation of the unit square by the linear transformation A
It is easy to see geometrically that
A[10]=[2203][10]=[20]=2[10].
It is less obvious (but easily checked once you find it) that
A[21]=[2203][21]=[63]=3[21].

Definition 5.3.5.

Let A∈Mn,n. An eigenvector for A is a vector x→∈Rn such that Ax→ is parallel to x→.
Figure 60. The map A stretches out the eigenvector [21] by a factor of 3 (the corresponding eigenvalue).
In other words, Ax→=λx→ for some scalar λ. If x→≠0→, then we say x→ is a nontrivial eigenvector and we call this λ an eigenvalue of A.

Activity 5.3.6.

Finding the eigenvalues λ that satisfy
Ax→=λx→=λ(Ix→)=(λI)x→
for some nontrivial eigenvector x→ is equivalent to finding nonzero solutions for the matrix equation
(A−λI)x→=0→.
(a)
If λ is an eigenvalue, and T is the transformation with standard matrix A−λI, which of these must contain a non-zero vector?
  1. The kernel of T
  2. The image of T
  3. The domain of T
  4. The codomain of T
(b)
Therefore, what can we conclude?
  1. A is invertible
  2. A is not invertible
  3. A−λI is invertible
  4. A−λI is not invertible

Definition 5.3.8.

The expression det(A−λI) is called the characteristic polynomial of A.
For example, when A=[1254], we have
A−λI=[1254]−[λ00λ]=[1−λ254−λ].
Thus the characteristic polynomial of A is
det[1−λ254−λ]=(1−λ)(4−λ)−(2)(5)=λ2−5λ−6
and its eigenvalues are the solutions −1,6 to λ2−5λ−6=0.

Activity 5.3.10.

Find all the eigenvalues for the matrix A=[3−32−4].

Activity 5.3.12.

Find all the eigenvalues for the matrix A=[3−310−42007].

Subsection 5.3.3 Individual Practice

Activity 5.3.13.

Let A∈Mn,n and λ∈R. The eigenvalues of A that correspond to λ are the vectors that get stretched by a factor of λ. Consider the following special cases for which we can make more geometric meaning.
(a)
What are some other ways we can think of the eigenvectors corresponding to eigenvalue λ=0?
(b)
What are some other ways we can think of the eigenvectors corresponding to eigenvalue λ=1?
(c)
What are some other ways we can think of the eigenvectors corresponding to eigenvalue λ=−1?

Subsection 5.3.4 Videos

Figure 61. Video: Finding eigenvalues

Exercises 5.3.5 Exercises

Subsection 5.3.6 Mathematical Writing Explorations

Exploration 5.3.14.

What are the maximum and minimum number of eigenvalues associated with an n×n matrix? Write small examples to convince yourself you are correct, and then prove this in generality.

Subsection 5.3.7 Sample Problem and Solution