TBIL-LA Eigenvalues and Characteristic Polynomials (GT3)

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Section 5.3 Eigenvalues and Characteristic Polynomials (GT3)

Learning Outcomes

Find the eigenvalues of a $2 \times 2$ matrix.

Subsection 5.3.1 Class Activities

Activity 5.3.1.

An invertible matrix $M$ and its inverse $M^{- 1}$ are given below:

M = [\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}] M^{- 1} = [\begin{array}{cc} - 2 & 1 \\ 3 / 2 & - 1 / 2 \end{array}]

Which of the following is equal to $det (M) det (M^{- 1}) ?$

$- 1$
$0$
$1$
$4$

Fact 5.3.2.

For every invertible matrix $M,$

det (M) det (M^{- 1}) = det (I) = 1

so $det (M^{- 1}) = \frac{1}{det (M)} .$

Furthermore, a square matrix $M$ is invertible if and only if $det (M) \neq 0 .$

Observation 5.3.3.

Consider the linear transformation $A : R^{2} \to R^{2}$ given by the matrix $A = [\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}] .$

Figure 65. Transformation of the unit square by the linear transformation $A$

It is easy to see geometrically that

A [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}] [\begin{matrix} 1 \\ 0 \end{matrix}] = [\begin{matrix} 2 \\ 0 \end{matrix}] = 2 [\begin{matrix} 1 \\ 0 \end{matrix}] .

It is less obvious (but easily checked once you find it) that

A [\begin{matrix} 2 \\ 1 \end{matrix}] = [\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}] [\begin{matrix} 2 \\ 1 \end{matrix}] = [\begin{matrix} 6 \\ 3 \end{matrix}] = 3 [\begin{matrix} 2 \\ 1 \end{matrix}] .

Definition 5.3.4.

Let $A \in M_{n, n} .$ An eigenvector for $A$ is a vector $\vec{x} \in R^{n}$ such that $A \vec{x}$ is parallel to $\vec{x} .$

Figure 66. The map $A$ stretches out the eigenvector $[\begin{matrix} 2 \\ 1 \end{matrix}]$ by a factor of $3$ (the corresponding eigenvalue).

In other words, $A \vec{x} = λ \vec{x}$ for some scalar $λ .$ If $\vec{x} \neq \vec{0},$ then we say $\vec{x}$ is a nontrivial eigenvector and we call this $λ$ an eigenvalue of $A .$

Activity 5.3.5.

Finding the eigenvalues $λ$ that satisfy

A \vec{x} = λ \vec{x} = λ (I \vec{x}) = (λ I) \vec{x}

for some nontrivial eigenvector $\vec{x}$ is equivalent to finding nonzero solutions for the matrix equation

(A - λ I) \vec{x} = \vec{0} .

Which of the following must be true for any eigenvalue?

The kernel of the transformation with standard matrix $A - λ I$ must contain the zero vector, so $A - λ I$ is invertible.
The kernel of the transformation with standard matrix $A - λ I$ must contain a non-zero vector, so $A - λ I$ is not invertible.
The image of the transformation with standard matrix $A - λ I$ must contain the zero vector, so $A - λ I$ is invertible.
The image of the transformation with standard matrix $A - λ I$ must contain a non-zero vector, so $A - λ I$ is not invertible.

Fact 5.3.6.

The eigenvalues $λ$ for a matrix $A$ are the values that make $A - λ I$ non-invertible.

Thus the eigenvalues $λ$ for a matrix $A$ are the solutions to the equation

det (A - λ I) = 0.

Definition 5.3.7.

The expression $det (A - λ I)$ is called characteristic polynomial of $A .$

For example, when $A = [\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}],$ we have

A - λ I = [\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}] - [\begin{array}{cc} λ & 0 \\ 0 & λ \end{array}] = [\begin{array}{cc} 1 - λ & 2 \\ 3 & 4 - λ \end{array}] .

Thus the characteristic polynomial of $A$ is

det [\begin{array}{cc} 1 - λ & 2 \\ 3 & 4 - λ \end{array}] = (1 - λ) (4 - λ) - (2) (3) = λ^{2} - 5 λ - 2

and its eigenvalues are the solutions to $λ^{2} - 5 λ - 2 = 0 .$

Activity 5.3.8.

Let $A = [\begin{array}{cc} 5 & 2 \\ - 3 & - 2 \end{array}] .$

(a)

Compute $det (A - λ I)$ to determine the characteristic polynomial of $A .$

(b)

Set this characteristic polynomial equal to zero and factor to determine the eigenvalues of $A .$

Activity 5.3.9.

Find all the eigenvalues for the matrix $A = [\begin{array}{cc} 3 & - 3 \\ 2 & - 4 \end{array}] .$

Activity 5.3.10.

Find all the eigenvalues for the matrix $A = [\begin{array}{cc} 1 & - 4 \\ 0 & 5 \end{array}] .$

Activity 5.3.11.

Find all the eigenvalues for the matrix $A = [\begin{array}{ccc} 3 & - 3 & 1 \\ 0 & - 4 & 2 \\ 0 & 0 & 7 \end{array}] .$

Subsection 5.3.2 Videos

Figure 67. Video: Finding eigenvalues

Subsection 5.3.3 Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/GT3.slides.html.

Exercises 5.3.4 Exercises

Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/GT3/.

Subsection 5.3.5 Mathematical Writing Explorations

Exploration 5.3.12.

What are the maximum and minimum number of eigenvalues associated with an

n \times n

matrix? Write small examples to convince yourself you are correct, and then prove this in generality.

Subsection 5.3.6 Sample Problem and Solution

Sample problem Example B.1.23.

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