Lesson test: Symmetric matrices and quadratic forms

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Symmetric matrices and quadratic forms

1.[1p]
A matrix is symmetric when

it equals its own transpose

Correct

its inverse equals its transpose

The answer is: it equals its own transpose

its determinant is one

The answer is: it equals its own transpose

its eigenvalues are distinct

The answer is: it equals its own transpose

2.[3p]
What does the spectral theorem guarantee for a real symmetric matrix?

Select all that apply

Its eigenvalues are all real

Correct

It has an orthonormal basis of eigenvectors

Correct

It can be written $A = Q D Q^{T}$ with $Q$ orthogonal

Correct

All its eigenvalues are positive

The answer is: Its eigenvalues are all real, It has an orthonormal basis of eigenvectors, It can be written $A = QDQ^T$ with $Q$ orthogonal

3.[3p]
Why are eigenvectors of distinct eigenvalues of a symmetric matrix perpendicular?

Computing $u^{T} A v$ two ways gives $λ (u \cdot v) = μ (u \cdot v)$ , forcing the dot product to zero

Correct

Because the determinant is positive

The answer is: Computing $\mathbf{u}^TA\mathbf{v}$ two ways gives $\lambda(\mathbf{u}\cdot\mathbf{v}) = \mu(\mathbf{u}\cdot\mathbf{v})$, forcing the dot product to zero

Because Gram-Schmidt was applied to them

The answer is: Computing $\mathbf{u}^TA\mathbf{v}$ two ways gives $\lambda(\mathbf{u}\cdot\mathbf{v}) = \mu(\mathbf{u}\cdot\mathbf{v})$, forcing the dot product to zero

Because symmetric matrices are always diagonal

The answer is: Computing $\mathbf{u}^TA\mathbf{v}$ two ways gives $\lambda(\mathbf{u}\cdot\mathbf{v}) = \mu(\mathbf{u}\cdot\mathbf{v})$, forcing the dot product to zero

4.[2p]
The form $5 x^{2} + 4 x y + 2 y^{2}$ has matrix $[(5, 2), (2, 2)]$ . What is its larger eigenvalue?

CorrectNot quite: 6

5.[3p]
The quadratic form $3 x^{2} + 2 x y + 3 y^{2}$ has eigenvalues 4 and 2. What is the shorter semi-axis of the curve where the form equals 1?

CorrectNot quite: 0.5

6.[3p]
The form $x^{2} + 4 x y + y^{2}$ has matrix $[(1, 2), (2, 1)]$ , with determinant $- 3$ . Its level curves are

hyperbolas, since the eigenvalues have opposite signs

Correct

ellipses, since the matrix is symmetric

The answer is: hyperbolas, since the eigenvalues have opposite signs

circles, since the diagonal entries are equal

The answer is: hyperbolas, since the eigenvalues have opposite signs

straight lines

The answer is: hyperbolas, since the eigenvalues have opposite signs

7.[2p]
For a two by two symmetric matrix, $a > 0$ together with a positive determinant means the form is positive definite.

True

Correct

False

The answer is: True

8.[3p]
Four points have covariance matrix $[(1.6667, 2), (2, 2.6667)]$ , with eigenvalues $4.2282$ and $0.1051$ . What percentage of the total variance lies along the first principal direction?

CorrectNot quite: 97.6

9.[3p]
Match each symmetric matrix to where it comes from.

$A^{T} A$
The Hessian
The covariance matrix
The inertia tensor

the spread of a cloud of data
the principal axes of a spinning body
the normal equations of a least squares fit
the second derivatives at a critical point

Show the answer

$A^{T} A$ : the normal equations of a least squares fit The Hessian: the second derivatives at a critical point The covariance matrix: the spread of a cloud of data The inertia tensor: the principal axes of a spinning body

1.[1p] A matrix is symmetric when

2.[3p] What does the spectral theorem guarantee for a real symmetric matrix?

3.[3p] Why are eigenvectors of distinct eigenvalues of a symmetric matrix perpendicular?

4.[2p] The form 5x2+4xy+2y2 has matrix [(5,2),(2,2)]. What is its larger eigenvalue?

5.[3p] The quadratic form 3x2+2xy+3y2 has eigenvalues 4 and 2. What is the shorter semi-axis of the curve where the form equals 1?

6.[3p] The form x2+4xy+y2 has matrix [(1,2),(2,1)], with determinant -3. Its level curves are

7.[2p] For a two by two symmetric matrix, a>0 together with a positive determinant means the form is positive definite.

8.[3p] Four points have covariance matrix [(1.6667,2),(2,2.6667)], with eigenvalues 4.2282 and 0.1051. What percentage of the total variance lies along the first principal direction?

9.[3p] Match each symmetric matrix to where it comes from.

1.[1p]
A matrix is symmetric when

2.[3p]
What does the spectral theorem guarantee for a real symmetric matrix?

3.[3p]
Why are eigenvectors of distinct eigenvalues of a symmetric matrix perpendicular?

4.[2p]
The form $5 x^{2} + 4 x y + 2 y^{2}$ has matrix $[(5, 2), (2, 2)]$ . What is its larger eigenvalue?

5.[3p]
The quadratic form $3 x^{2} + 2 x y + 3 y^{2}$ has eigenvalues 4 and 2. What is the shorter semi-axis of the curve where the form equals 1?

6.[3p]
The form $x^{2} + 4 x y + y^{2}$ has matrix $[(1, 2), (2, 1)]$ , with determinant $- 3$ . Its level curves are

7.[2p]
For a two by two symmetric matrix, $a > 0$ together with a positive determinant means the form is positive definite.

8.[3p]
Four points have covariance matrix $[(1.6667, 2), (2, 2.6667)]$ , with eigenvalues $4.2282$ and $0.1051$ . What percentage of the total variance lies along the first principal direction?

9.[3p]
Match each symmetric matrix to where it comes from.