TBIL-LA Eigenvalues and Characteristic Polynomials (GT3)

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Section 5.3 Eigenvalues and Characteristic Polynomials (GT3)

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Learning Outcomes

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

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Subsection 5.3.1 Warm Up

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Activity 5.3.1.

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Let

R : R^{2} \to R^{2}

be the transformation given by rotating vectors about the origin through and angle of

45^{\circ},

and let

S : R^{2} \to R^{2}

denote the transformation that reflects vectors about the line

x_{1} = x_{2} .

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(a)

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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 ?

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(b)

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Now consider the transformation

S .

Are there any lines

L

for which

S (L)

is parallel to

L ?

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Subsection 5.3.2 Class Activities

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Activity 5.3.2.

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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}]

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Which of the following is equal to

det (M) det (M^{- 1}) ?

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

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Fact 5.3.3.

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For every invertible matrix

M,

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

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

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Furthermore, a square matrix

M

is invertible if and only if

det (M) \neq 0 .

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Observation 5.3.4.

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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 59. Transformation of the unit square by the linear transformation $A$

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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}] .

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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}] .

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Definition 5.3.5.

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

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In other words,

A \vec{x} = λ \vec{x}

for some scalar

λ .

\vec{x} \neq \vec{0},

then we say

\vec{x}

is a nontrivial eigenvector and we call this

λ

an eigenvalue of

A .

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Activity 5.3.6.

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Finding the eigenvalues

λ

that satisfy

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

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for some nontrivial eigenvector

\vec{x}

is equivalent to finding nonzero solutions for the matrix equation

(A - λ I) \vec{x} = \vec{0} .

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(a)

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λ

is an eigenvalue, and

T

is the transformation with standard matrix

A - λ I,

which of these must contain a non-zero vector?

The kernel of $T$
The image of $T$
The domain of $T$
The codomain of $T$

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(b)

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Therefore, what can we conclude?

$A$ is invertible
$A$ is not invertible
$A - λ I$ is invertible
$A - λ I$ is not invertible

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(c)

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And what else?

$det A = 0$
$det A = 1$
$det (A - λ I) = 0$
$det (A - λ I) = 1$

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Fact 5.3.7.

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The eigenvalues

λ

for a matrix

A

are exactly the values that make

A - λ I

non-invertible.

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Thus the eigenvalues

λ

for a matrix

A

are the solutions to the equation

det (A - λ I) = 0.

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Definition 5.3.8.

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The expression

det (A - λ I)

is called the characteristic polynomial of

A .

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For example, when

A = [\begin{array}{cc} 1 & 2 \\ 5 & 4 \end{array}],

we have

A - λ I = [\begin{array}{cc} 1 & 2 \\ 5 & 4 \end{array}] - [\begin{array}{cc} λ & 0 \\ 0 & λ \end{array}] = [\begin{array}{cc} 1 - λ & 2 \\ 5 & 4 - λ \end{array}] .

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Thus the characteristic polynomial of

A

det [\begin{array}{cc} 1 - λ & 2 \\ 5 & 4 - λ \end{array}] = (1 - λ) (4 - λ) - (2) (5) = λ^{2} - 5 λ - 6

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and its eigenvalues are the solutions

- 1, 6

λ^{2} - 5 λ - 6 = 0 .

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Activity 5.3.9.

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Let

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

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(a)

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Compute

det (A - λ I)

to determine the characteristic polynomial of

A .

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(b)

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

A .

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Activity 5.3.10.

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Find all the eigenvalues for the matrix

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

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Activity 5.3.11.

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Find all the eigenvalues for the matrix

A = [\begin{array}{cc} 1 & - 4 \\ 0 & 5 \end{array}] .

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Activity 5.3.12.

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Find all the eigenvalues for the matrix

A = [\begin{array}{ccc} 3 & - 3 & 1 \\ 0 & - 4 & 2 \\ 0 & 0 & 7 \end{array}] .

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Subsection 5.3.3 Individual Practice

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Activity 5.3.13.

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Let

A \in M_{n, n}

and

λ \in 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.

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(a)

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What are some other ways we can think of the eigenvalues corresponding to eigenvalue

λ = 0 ?

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(b)

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What are some other ways we can think of the eigenvalues corresponding to eigenvalue

λ = 1 ?

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(c)

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What are some other ways we can think of the eigenvalues corresponding to eigenvalue

λ = - 1 ?

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(d)

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How might we interpret a matrix that has no (real) eigenvectors/values?

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Subsection 5.3.4 Videos

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Figure 61. Video: Finding eigenvalues

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Exercises 5.3.5 Exercises

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Exercises available at https://tbil.org/linear-algebra/preview/exercises/#/bank/GT3/.

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Subsection 5.3.6 Mathematical Writing Explorations

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Exploration 5.3.14.

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.

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Subsection 5.3.7 Sample Problem and Solution

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Sample problem Example B.1.24.

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