Q: If $$A is an$$ $$m \\times n$$ matrix, which of the following is always true about the matrices $$U$$, $$\\Sigma$$, and $$V in $$the SVD $$A = U \\Sigma V^T$$?

$$U is$$ $$m \\times m$$, $$\\Sigma is$$ $$m \\times n$$, $$V is$$ $$n \\times n.$$

Q: Which property do the columns of $$U$$ and $$V$$ have in the SVD $$A = U \\Sigma V^T$$?

They form orthonormal bases for $$\\mathbb{R}^m$$ and $$\\mathbb{R}^n$$, respectively.

Q: If $$A is a$$ $$2 \\times 2$$ matrix with $$A^TA = \\begin{pmatrix} 5 & 2 \\ 2 & 5 \\end{pmatrix}$$, what are the singular values of $$A$$?

$$\\sqrt{7}$$ and $$\\sqrt{3}$$

Q: Which of the following is NOT a property of the SVD?

The matrix $$\\Sigma is $$always square.

Question 1

Which of the following best describes the Singular Value Decomposition (SVD) of a matrix $$A$$?

Accepted Answer

It expresses $$A as a $$product of three matrices: $$A = U \Sigma V^T$$, where $$U$$ and $$V$$ are orthogonal and $$\Sigma is $$diagonal.

Question 2

Given a matrix $$A$$, what do the nonzero singular values of $$A$$ represent?

Accepted Answer

The square roots of the nonzero eigenvalues of $$A^TA.$$

Question 3

If $$A is an$$ $$m \times n$$ matrix, which of the following is always true about the matrices $$U$$, $$\Sigma$$, and $$V in $$the SVD $$A = U \Sigma V^T$$?

Accepted Answer

$$U is$$ $$m \times m$$, $$\Sigma is$$ $$m \times n$$, $$V is$$ $$n \times n.$$

Question 4

Which property do the columns of $$U$$ and $$V$$ have in the SVD $$A = U \Sigma V^T$$?

Accepted Answer

They form orthonormal bases for $$\mathbb{R}^m$$ and $$\mathbb{R}^n$$, respectively.

Question 5

Suppose $$A is a$$ $$3 \times 2$$ matrix with rank 2. How many nonzero singular values does $$A$$ have?

Accepted Answer

2

Question 6

Which of the following statements is true about the relationship between the singular values of $$A$$ and the eigenvalues of $$A^TA$$?

Accepted Answer

The singular values of $$A$$ are the square roots of the eigenvalues of $$A^TA.$$

Question 7

In the SVD $$A = U \Sigma V^T$$, what does the matrix $$\Sigma$$ contain?

Accepted Answer

The singular values of $$A on $$the diagonal and zeros elsewhere.

Question 8

Which of the following is a correct application of SVD in real-world scenarios?

Accepted Answer

Image compression by keeping only the largest singular values.

Question 9

If $$A is a$$ $$4 \times 4$$ matrix with singular values $$5, 3, 1, 0$$, what is the rank of $$A$$?

Accepted Answer

3

Question 10

Which step is necessary to compute the SVD of a matrix $$A$$?

Accepted Answer

Find the eigenvalues and eigenvectors of $$A^TA$$ and $$AA^T.$$

Question 11

Given $$A = U \Sigma V^T$$, which matrix is always diagonal?

Accepted Answer

$$\Sigma$$

Question 12

If $$A is a$$ $$2 \times 2$$ matrix with $$A^TA = \begin{pmatrix} 5 & 2 \ 2 & 5 \end{pmatrix}$$, what are the singular values of $$A$$?

Accepted Answer

$$\sqrt{7}$$ and $$\sqrt{3}$$

Question 13

Which of the following is NOT a property of the SVD?

Accepted Answer

The matrix $$\Sigma is $$always square.

Question 14

What is the geometric interpretation of the largest singular value of $$A$$?

Accepted Answer

It is the length of the longest axis of the ellipse formed by transforming the unit circle under $$A.$$

Question 15

If $$A is a$$ $$3 \times 3$$ matrix and $$A^TA$$ has eigenvalues $$9, 4, 1$$, what are the singular values of $$A$$?

Accepted Answer

$$3, 2, 1$$

Question 16

Which of the following is a consequence of the SVD for any $$m \times n$$ matrix $$A$$?

Accepted Answer

Any matrix can be written as a sum of rank-1 matrices weighted by singular values.

Question 17

Suppose $$A is a$$ $$5 \times 3$$ matrix with rank 2. What is the maximum number of nonzero singular values $$A$$ can have?

Accepted Answer

2

Question 18

Which of the following is true about the stability of singular values compared to eigenvalues?

Accepted Answer

Singular values are generally more stable under small perturbations of the matrix than eigenvalues.

Question 19

If $$A is a$$ $$2 \times 2$$ matrix with SVD $$A = U \Sigma V^T$$, and $$\Sigma = \begin{pmatrix} 4 & 0 \ 0 & 2 \end{pmatrix}$$, what is $$\|A\|_2 ($$the operator norm)?

Accepted Answer

$$4$$

Question 20

Which of the following is a correct statement about the SVD and the rank of a matrix?

Accepted Answer

The rank of $$A is $$equal to the number of nonzero singular values.

Singular Value Decomposition (SVD) and Its Applications in Linear Algebra

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