Deck 6: Eigenvalues and Eigenvectors

Question 1

Determine which of , , and are eigenvectors of , and determine the associated eigenvalues.

Answer 1

is an eigenvector with eigenvalue 2,

is an eigenvector with eigenvalue



.

Question 2

Determine which of , , and are eigenvectors of
,
and determine the associated eigenvalues.

Answer 2

is an eigenvector with eigenvalue 0,

is an eigenvector with eigenvalue 2.

Question 3

Find a basis for the eigenspace associated with eigenvalue for matrix
.

Answer 3

Question 4

Find a basis for the eigenspace associated with eigenvalue for matrix
.

Question 5

Find a basis for the eigenspace associated with eigenvalue for the matrix

.

Question 6

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A =
.

Question 7

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A =
.

Question 8

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A =
.

Question 9

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A =
.

Question 10

Find the characteristic polynomial, the eigenvalues, and a basis for each eigenspace for the matrix A =
.

Question 11

An matrix A can have no more than n eigenvalues.

Question 12

If is the characteristic polynomial of an matrix A, and , then A is not invertible.

Question 13

Suppose the matrix A has n distinct eigenvalues. Then the dimension of each eigenspace is 1.

Question 14

If u and v are both eigenvectors of an n ×n matrix A, then u+v is also an eigenvector of the A.

Question 15

If is an eigenvalue of an invertible matrix A, then is an eigenvalue of the matrix
.

Question 16

Compute if
.

Question 17

Compute if
.

Question 18

Find the matrix A that has the given eigenvalues and corresponding eigenvectors.

Question 19

Find the matrix A that has the given eigenvalues and bases for the corresponding eigenspaces.

;
;

Question 20

Find the matrix A that has the given eigenvalues and bases for the corresponding eigenspaces.

Question 21

Diagonalize the matrix A, if possible.

Question 22

Diagonalize the matrix A, if possible.

Question 23

Diagonalize the matrix A, if possible.

Question 24

Diagonalize the given matrix A, and use the diagonalization to compute
.

Question 25

Diagonalize the given matrix A, and use the diagonalization to compute
.

Question 26

If an matrix A has n distinct eigenvalues, then A is diagonalizable.

Question 27

The matrix is diagonalizable.

Question 28

If A and are diagonalizable matrices, then AB is diagonalizable.

Question 29

If and , where , and and are nonzero vectors, then is linearly independent.

Question 30

If A is diagonalizable, then is diagonalizable.

Question 31

Find the eigenvalues and a basis for each eigenspace for the given matrix.

Question 32

Find the eigenvalues and a basis for each eigenspace for the given matrix.

Question 33

Find the eigenvalues and a basis for each eigenspace for the given matrix.

Question 34

Find the eigenvalues and a basis for each eigenspace for the given matrix.

Question 35

Determine the rotation and dilation that result from multiplying vectors in by the given matrix.

Question 36

Determine the rotation and dilation that result from multiplying vectors in by the given matrix.

Question 37

Find the rotation-dilation matrix B within the given matrix A.

Question 38

Find the rotation-dilation matrix B within the given matrix A.

Question 39

Factor the matrix from Question 1 in the form where B is a rotation-dilation matrix.

Question 40

Factor the matrix from Question 2 in the form where B is a rotation-dilation matrix.

Question 41

Factor the given matrix A in the form where B is a rotation-dilation matrix. Find the dilation and angle of rotation. Use this information to evaluate the matrix power without computing it directly.

Question 42

If A is a real matrix, and is a complex eigenvalue of A, then is also an eigenvalue of A.

Question 43

If A is a real matrix and is an eigenvalue of A with and corresponding eigenvector u, then
.

Question 44

If the invertible matrix A has hidden rotation-dilation matrix , where then has hidden rotation-dilation matrix
.

Question 45

If is an eigenvalue of the real matrix A with corresponding eigenvector , then is an eigenvalue of with corresponding eigenvector
.

Question 46

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Question 47

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Question 48

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Question 49

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Question 50

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Question 51

The coefficient matrix for a system of linear differential equations of the form has the given eigenvalues and eigenspace bases. Find the general solution for the system.

Question 52

Find the general solution for the system
.

Question 53

Find the general solution for the system
.

Question 54

Find the solution for the system that satisfies the condition at t = 0.

Question 55

Find the solution for the system that satisfies the condition at t = 0.​

Question 56

Suppose that two countries are in an arms race modeled by the system of differential equations

where y₁ and y₂ are measured in thousands. Find the solution for the system with initial conditions , and use it to predict the long-term amounts of arms held by each country.

Question 57

If , and are solutions to the initial-value problem , with , then for all t.

Question 58

If A is a real square matrix with complex eigenvalue and associated eigenvector , then is a real solution to the system
.

Question 59

Suppose that A is an matrix and is a solution to the system of linear differential equations where is an eigenvector of A with associated eigenvalue
. Let k be any scalar. Then is a solution to the system
.

Question 60

Suppose that A is an matrix and is a solution to the system of linear differential equations where is an eigenvector of A with associated eigenvalue
. If A is invertible, then is a solution to the system
.

Question 61

Compute the first three iterations of the Power Method without scaling, starting with the given , where
.

Question 62

Compute the first three iterations of the Power Method without scaling, starting with the given , where
.

Question 63

Compute the first two iterations of the Power Method with scaling, starting with the given , rounding any numerical values to two decimal places.

Question 64

Compute the first two iterations of the Power Method with scaling, starting with the given , rounding any numerical values to two decimal places.

Question 65

Compute the first two iterations of the Inverse Power Method, starting with the given , rounding any numerical values to two decimal places.

Question 66

Compute the first two iterations of the Shifted Inverse Power Method, starting with the given , to determine the eigenvalue of A closest to , rounding any numerical values to two decimal places.

Question 67

The dominant eigenvalue of the matrix A given below is
. Compute the first two iterations of the Shifted Power Method with scaling, starting with the given , to determine the eigenvalue farthest from , rounding any numerical values to two decimal places.

Question 68

Use the Power Method with scaling to determine an eigenvalue and associated eigenvector of A, starting with the given
.

Question 69

Use the Power Method with scaling to determine an eigenvalue and associated eigenvector of A, starting with the given
.

Question 70

Use the Power Method with scaling to determine an eigenvalue and associated eigenvector of A, starting with the given
.

Question 71

The Power Method applied to the matrix and vector converges.

Question 72

The Power Method applied to the matrix and vector converges.

Question 73

The Inverse Power Method applied to the matrix and vector converges.

Question 74

The Shifted Inverse Power Method for an invertible n×nmatrix Ais implemented by applying the Power Method to for some scalar c.

Question 75

Suppose is a matrix having eigenvalues
. Then has dominant eigenvalue
.