Linear Algebra and Its Applications

David C. Lay, Steven R. Lay, Judi J. McDonald

Chapter 5

Eigenvalues and Eigenvectors - all with Video Answers

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Section 1

Eigenvectors and Eigenvalues

Problem 1

Is $\lambda=2$ an eigenvalue of $\left[\begin{array}{ll}{3} & {2} \\ {3} & {8}\end{array}\right] ?$ Why or why not?

Rakvi ..

Rakvi ..

Numerade Educator

Problem 2

Is $\lambda=-2$ an eigenvalue of $\left[\begin{array}{rr}{7} & {3} \\ {3} & {-1}\end{array}\right] ?$ Why or why not?

Amy J.

Numerade Educator

Problem 3

Is $\left[\begin{array}{l}{1} \\ {4}\end{array}\right]$ an eigenvector of $\left[\begin{array}{ll}{-3} & {1} \\ {-3} & {8}\end{array}\right] ?$ If so, find the eigenvalue.

Rakvi ..

Numerade Educator

Problem 4

Is $\left[\begin{array}{c}{-1+\sqrt{2}} \\ {1}\end{array}\right]$ an eigenvector of $\left[\begin{array}{ll}{2} & {1} \\ {1} & {4}\end{array}\right] ?$ If so, find the eigenvalue.

Amy J.

Numerade Educator

Problem 5

Is $\left[\begin{array}{r}{4} \\ {-3} \\ {1}\end{array}\right]$ an eigenvector of $\left[\begin{array}{rrr}{3} & {7} & {9} \\ {-4} & {-5} & {1} \\ {2} & {4} & {4}\end{array}\right] ?$ If so, find the eigenvalue.

Rakvi ..

Numerade Educator

Problem 6

Is $\left[\begin{array}{r}{1} \\ {-2} \\ {1}\end{array}\right]$ an eigenvector of $\left[\begin{array}{ccc}{3} & {6} & {7} \\ {3} & {3} & {7} \\ {5} & {6} & {5}\end{array}\right] ?$ If so, find the eigenvalue.

Amy J.

Numerade Educator

Problem 7

Is $\lambda=4$ an eigenvalue of $\left[\begin{array}{rrr}{3} & {0} & {-1} \\ {2} & {3} & {1} \\ {-3} & {4} & {5}\end{array}\right] ?$ If so, find one corresponding eigenvector.

Rakvi ..

Numerade Educator

Problem 8

Is $\lambda=3$ an eigenvalue of $\left[\begin{array}{rrr}{1} & {2} & {2} \\ {3} & {-2} & {1} \\ {0} & {1} & {1}\end{array}\right] ?$ If so, find one corresponding eigenvector.

Amy J.

Numerade Educator

Problem 9

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{ll}{5} & {0} \\ {2} & {1}\end{array}\right], \lambda=1,5
$$

Dharmendra J.

Numerade Educator

Problem 10

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{rr}{10} & {-9} \\ {4} & {-2}\end{array}\right], \lambda=4
$$

Amy J.

Numerade Educator

Problem 11

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{rr}{4} & {-2} \\ {-3} & {9}\end{array}\right], \lambda=10
$$

Jim V.

Numerade Educator

Problem 12

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{rr}{7} & {4} \\ {-3} & {-1}\end{array}\right], \lambda=1,5
$$

Amy J.

Numerade Educator

Problem 13

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{rrr}{4} & {0} & {1} \\ {-2} & {1} & {0} \\ {-2} & {0} & {1}\end{array}\right], \lambda=1,2,3
$$

Rakvi ..

Numerade Educator

Problem 14

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{rrr}{1} & {0} & {-1} \\ {1} & {-3} & {0} \\ {4} & {-13} & {1}\end{array}\right], \lambda=-2
$$

Amy J.

Numerade Educator

Problem 15

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{rrr}{4} & {2} & {3} \\ {-1} & {1} & {-3} \\ {2} & {4} & {9}\end{array}\right], \lambda=3
$$

Dharmendra J.

Numerade Educator

Problem 16

In Exercises $9-16,$ find a basis for the eigenspace corresponding to each listed eigenvalue.
$$
A=\left[\begin{array}{cccc}{3} & {0} & {2} & {0} \\ {1} & {3} & {1} & {0} \\ {0} & {1} & {1} & {0} \\ {0} & {0} & {0} & {4}\end{array}\right], \lambda=4
$$

Amy J.

Numerade Educator

Problem 17

Find the eigenvalues of the matrices in Exercises 17 and 18 .
$$
\left[\begin{array}{rrr}{0} & {0} & {0} \\ {0} & {2} & {5} \\ {0} & {0} & {-1}\end{array}\right]
$$

Dharmendra J.

Numerade Educator

Problem 18

Find the eigenvalues of the matrices in Exercises 17 and 18 .
$$
\left[\begin{array}{ccc}{4} & {0} & {0} \\ {0} & {0} & {0} \\ {1} & {0} & {-3}\end{array}\right]
$$

Gennady N.

Numerade Educator

Problem 19

$\operatorname{For} A=\left[\begin{array}{ccc}{1} & {2} & {3} \\ {1} & {2} & {3} \\ {1} & {2} & {3}\end{array}\right],$ find one eigenvalue, with no calculation. Justify your answer.

EL

Elias L.

Numerade Educator

Problem 20

Without calculation, find one eigenvalue and two linearly independent eigenvectors of $A=\left[\begin{array}{ccc}{5} & {5} & {5} \\ {5} & {5} & {5} \\ {5} & {5} & {5}\end{array}\right] .$ Justify your answer.

Gennady N.

Numerade Educator

Problem 21

In Exercises 21 and $22, A$ is an $n \times n$ matrix. Mark each statement True or False. Justify each answer.
a. If $A \mathbf{x}=\lambda \mathbf{x}$ for some vector $\mathbf{x},$ then $\lambda$ is an eigenvalue of $A .$
b. A matrix $A$ is not invertible if and only if 0 is an eigen- value of $A .$
c. A number $c$ is an eigenvalue of $A$ if and only if the equation $(A-c I) \mathbf{x}=\mathbf{0}$ has a nontrivial solution.
d. Finding an eigenvector of $A$ may be difficult, but checking whether a given vector is in fact an eigenvector is easy.
e. To find the eigenvalues of $A,$ reduce $A$ to echelon form.

Dharmendra J.

Numerade Educator

Problem 22

In Exercises 21 and $22, A$ is an $n \times n$ matrix. Mark each statement True or False. Justify each answer.
a. If $A \mathbf{x}=\lambda \mathbf{x}$ for some scalar $\lambda,$ then $\mathbf{x}$ is an eigenvector of $A .$
b. If $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ are linearly independent eigenvectors, then they correspond to distinct eigenvalues.
c. A steady-state vector for a stochastic matrix is actually an eigenvector.
d. The eigenvalues of a matrix are on its main diagonal.
e. An eigenspace of $A$ is a null space of a certain matrix.

Dharmendra J.

Numerade Educator

Problem 23

Explain why a $2 \times 2$ matrix can have at most two distinct eigenvalues. Explain why an $n \times n$ matrix can have at most $n$ distinct eigenvalues.

Rakvi ..

Numerade Educator

Problem 24

Construct an example of a $2 \times 2$ matrix with only one distinct eigenvalue.

Gennady N.

Numerade Educator

Problem 25

Let $\lambda$ be an eigenvalue of an invertible matrix $A .$ Show that $\lambda^{-1}$ is an eigenvalue of $A^{-1} .[\text { Hint: Suppose a nonzero } \mathbf{x}$ satisfies $A \mathbf{x}=\lambda \mathbf{x} . ]$

Dharmendra J.

Numerade Educator

Problem 26

Show that if $A^{2}$ is the zero matrix, then the only eigenvalue of $A$ is $0 .$

Gennady N.

Numerade Educator

Problem 27

Show that $\lambda$ is an eigenvalue of $A$ if and only if $\lambda$ is an eigenvalue of $A^{T} .\left[\text { Hint: Find out how } A-\lambda I \text { and } A^{T}-\lambda I\right.$ are related.

Dharmendra J.

Numerade Educator

Problem 28

Use Exercise 27 to complete the proof of Theorem 1 for the case when $A$ is lower triangular.

Dharmendra J.

Numerade Educator

Problem 29

Consider an $n \times n$ matrix $A$ with the property that the row sums all equal the same number $s .$ Show that $s$ is an eigenvalue of $A .[\text { Hint: Find an eigenvector. }]$

Dharmendra J.

Numerade Educator

Problem 30

Consider an $n \times n$ matrix $A$ with the property that the column sums all equal the same number $s$ . Show that $s$ is an eigenvalue of $A .[\text { Hint: Use Exercises } 27 \text { and } 29 .]$

Dharmendra J.

Numerade Educator

Problem 31

In Exercises 31 and $32,$ let $A$ be the matrix of the linear transformation $T .$ Without writing $A$ , find an eigenvalue of $A$ and describe the eigenspace.
$T$ is the transformation on $\mathbb{R}^{2}$ that reflects points across some line through the origin.

Dharmendra J.

Numerade Educator

Problem 32

In Exercises 31 and $32,$ let $A$ be the matrix of the linear transformation $T .$ Without writing $A$ , find an eigenvalue of $A$ and describe the eigenspace.
$T$ is the transformation on $\mathbb{R}^{3}$ that rotates points about some line through the origin.

Dharmendra J.

Numerade Educator

Problem 33

Let $\mathbf{u}$ and $\mathbf{v}$ be eigenvectors of a matrix $A,$ with corresponding eigenvalues $\lambda$ and $\mu,$ and let $c_{1}$ and $c_{2}$ be scalars. Define $\mathbf{x}_{k}=c_{1} \lambda^{k} \mathbf{u}+c_{2} \mu^{k} \mathbf{v} \quad(k=0,1,2, \ldots)$
a. What is $\mathbf{x}_{k+1},$ by definition?
b. Compute $A \mathbf{x}_{k}$ from the formula for $\mathbf{x}_{k},$ and show that $A \mathbf{x}_{k}=\mathbf{x}_{k+1} .$ This calculation will prove that the sequence $\left\{\mathbf{x}_{k}\right\}$ defined above satisfies the difference equation $\mathbf{x}_{k+1}=A \mathbf{x}_{k}(k=0,1,2, \ldots)$

Dharmendra J.

Numerade Educator

Problem 34

Describe how you might try to build a solution of a difference equation $\mathbf{x}_{k+1}=A \mathbf{x}_{k}(k=0,1,2, \ldots)$ if you were given the initial $\mathbf{x}_{0}$ and this vector did not happen to be an eigenvector of $A \cdot\left[\text { Hint: How might you relate } \mathbf{x}_{0} \text { to eigenvectors of } A ?\right]$

Dharmendra J.

Numerade Educator

Problem 35

Let $\mathbf{u}$ and $\mathbf{v}$ be the vectors shown in the figure, and suppose $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors of a $2 \times 2$ matrix $A$ that correspond to eigenvalues 2 and $3,$ respectively. Let $T : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be the linear transformation given by $T(\mathbf{x})=A \mathbf{x}$ for each $\mathbf{x}$ in $\mathbb{R}^{2},$ and let $\mathbf{w}=\mathbf{u}+\mathbf{v} .$ Make a copy of the figure, and on the same coordinate system, carefully plot the vectors $T(\mathbf{u})$ $T(\mathbf{v}),$ and $T(\mathbf{w})$
FIGURE NOT COPY

Dharmendra J.

Numerade Educator

Problem 36

Repeat Exercise $35,$ assuming $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors of $A$ that correspond to eigenvalues $-1$ and $3,$ respectively.

Gennady N.

Numerade Educator

Problem 37

In Exercises $37-40$ , use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.
$$
\left[\begin{array}{rrr}{8} & {-10} & {-5} \\ {2} & {17} & {2} \\ {-9} & {-18} & {4}\end{array}\right]
$$

Check back soon!

Problem 38

In Exercises $37-40$ , use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.
$$
\left[\begin{array}{rrrr}{9} & {-4} & {-2} & {-4} \\ {-56} & {32} & {-28} & {44} \\ {-14} & {-14} & {6} & {-14} \\ {42} & {-33} & {21} & {-45}\end{array}\right]
$$

Gennady N.

Numerade Educator

Problem 39

In Exercises $37-40$ , use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.
$$
\left[\begin{array}{rrrrr}{4} & {-9} & {-7} & {8} & {2} \\ {-7} & {-9} & {0} & {7} & {14} \\ {5} & {10} & {5} & {-5} & {-10} \\ {-2} & {3} & {7} & {0} & {4} \\ {-3} & {-13} & {-7} & {10} & {11}\end{array}\right]
$$

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Problem 40

In Exercises $37-40$ , use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.
$$
\left[\begin{array}{ccccc}{-4} & {-4} & {20} & {-8} & {-1} \\ {14} & {12} & {46} & {18} & {2} \\ {6} & {4} & {-18} & {8} & {1} \\ {11} & {7} & {-37} & {17} & {2} \\ {18} & {12} & {-60} & {24} & {5}\end{array}\right]
$$

Check back soon!