Lesson test: Eigenvectors and eigenvalues

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Eigenvectors and eigenvalues

1.[2p]
A vector $v$ is an eigenvector of $A$ when

$v$ is non-zero and $A v = λ v$ for some scalar $λ$

Correct

$A v = 0$

The answer is: $\mathbf{v}$ is non-zero and $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$

$v$ has length one and $A v$ has length one

The answer is: $\mathbf{v}$ is non-zero and $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$

$A v$ is perpendicular to $v$

The answer is: $\mathbf{v}$ is non-zero and $A\mathbf{v} = \lambda\mathbf{v}$ for some scalar $\lambda$

2.[2p]
Eigenvalues are found by solving

$\det (A - λ I) = 0$

Correct

$\det (A) = λ$

The answer is: $\det(A - \lambda I) = 0$

$A x = b$

The answer is: $\det(A - \lambda I) = 0$

$A^{T} A = λ I$

The answer is: $\det(A - \lambda I) = 0$

3.[2p]
$A = [(4, 1), (- 2, 1)]$ has eigenvalues $2$ and what other value?

CorrectNot quite: 3

4.[3p]
A two by two matrix has trace $7$ and determinant $12$ . What is its larger eigenvalue?

CorrectNot quite: 4

5.[2p]
A matrix is singular exactly when one of its eigenvalues is zero.

True

Correct

False

The answer is: True

6.[3p]
Which of these are true when $A v = λ v$ with $A$ invertible?

Select all that apply

$A^{k} v = λ^{k} v$

Correct

$A^{- 1} v = (1 / λ) v$

Correct

$(A + c I) v = (λ + c) v$

Correct

$A v$ is perpendicular to $v$

The answer is: $A^k\mathbf{v} = \lambda^k\mathbf{v}$, $A^{-1}\mathbf{v} = (1/\lambda)\mathbf{v}$, $(A + cI)\mathbf{v} = (\lambda + c)\mathbf{v}$

7.[2p]
Why does a quarter turn have no real eigenvector?

It moves every direction, and its characteristic equation $λ^{2} + 1 = 0$ has no real root

Correct

Its determinant is zero

The answer is: It moves every direction, and its characteristic equation $\lambda^2 + 1 = 0$ has no real root

Its trace is one

The answer is: It moves every direction, and its characteristic equation $\lambda^2 + 1 = 0$ has no real root

Rotations are not linear

The answer is: It moves every direction, and its characteristic equation $\lambda^2 + 1 = 0$ has no real root

8.[3p]
The shear $[(1, 0), (1, 1)]$ has $λ = 1$ repeated. How many independent eigenvectors does it have?

CorrectNot quite: 1

9.[3p]
Match each matrix to its eigenvalues.

A projection onto a line
A reflection in a line
Scaling everything by 5
Triangular with diagonal entries 2, 3, 7

2, 3 and 7
5 and 5
1 and 0
1 and -1

Show the answer

A projection onto a line: 1 and 0 A reflection in a line: 1 and -1 Scaling everything by 5: 5 and 5 Triangular with diagonal entries 2, 3, 7: 2, 3 and 7

1.[2p] A vector v is an eigenvector of A when

2.[2p] Eigenvalues are found by solving

3.[2p] A=[(4,1),(-2,1)] has eigenvalues 2 and what other value?

4.[3p] A two by two matrix has trace 7 and determinant 12. What is its larger eigenvalue?

5.[2p] A matrix is singular exactly when one of its eigenvalues is zero.

6.[3p] Which of these are true when Av=λv with A invertible?

7.[2p] Why does a quarter turn have no real eigenvector?

8.[3p] The shear [(1,0),(1,1)] has λ=1 repeated. How many independent eigenvectors does it have?

9.[3p] Match each matrix to its eigenvalues.

1.[2p]
A vector $v$ is an eigenvector of $A$ when

2.[2p]
Eigenvalues are found by solving

3.[2p]
$A = [(4, 1), (- 2, 1)]$ has eigenvalues $2$ and what other value?

4.[3p]
A two by two matrix has trace $7$ and determinant $12$ . What is its larger eigenvalue?

5.[2p]
A matrix is singular exactly when one of its eigenvalues is zero.

6.[3p]
Which of these are true when $A v = λ v$ with $A$ invertible?

7.[2p]
Why does a quarter turn have no real eigenvector?

8.[3p]
The shear $[(1, 0), (1, 1)]$ has $λ = 1$ repeated. How many independent eigenvectors does it have?

9.[3p]
Match each matrix to its eigenvalues.