Q: Suppose $$A is a$$ $$3 \times 2$$ matrix. Which of the following statements about $$A^T A is $$true?

$$A^T A is a$$ $$2 \times 2$$ symmetric matrix$

Question 1

Which of the following best describes the singular value decomposition (SVD) of an $$m 	imes n$$ matrix $$A$$?

Accepted Answer

$$A = U \Sigma V^T$$, where $$U$$ and $$V$$ are orthogonal matrices and $$\Sigma is a $$diagonal matrix with non-negative real numbers$

Question 2

Given a $$3 	imes 2$$ matrix $$A$$, what is the maximum possible number of nonzero singular values of $$A$$?

Accepted Answer

$$2$$

Question 3

If $$A is a$$ $$4 	imes 4$$ symmetric matrix, which of the following statements about its singular values is true?

Accepted Answer

The singular values of $$A$$ are the absolute values of its eigenvalues

Question 4

Suppose $$A is a$$ $$2 	imes 2$$ matrix with $$A^T A = \begin{pmatrix} 5 & 2 \ 2 & 5 \end{pmatrix}. $$What is the sum of the squares of the singular values of $$A$$?

Accepted Answer

$$10$$

Question 5

Which of the following is a correct interpretation of the rank of a matrix $$A in $$terms of its singular values?

Accepted Answer

The rank of $$A$$ equals the number of nonzero singular values of $$A$$

Question 6

Let $$A be a$$ $$3 	imes 3$$ matrix with singular values $$\$$sigma_1$$ = 4$$, $$\$$sigma_2$$ = 2$$, $$\$$sigma_3$$ = 0. $$What is the rank of $$A$$?

Accepted Answer

$$2$$

Question 7

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

Accepted Answer

$$U$$ and $$V$$ are orthogonal matrices$

Question 8

Which of the following is a real-world application of singular value decomposition (SVD)?

Accepted Answer

Image compression and noise reduction

Question 9

Given $$A is a$$ $$2 	imes 2$$ matrix with $$A^T A$$ having eigenvalues $$9$$ and $$4$$, what are the singular values of $$A$$?

Accepted Answer

$$3$$ and $$2$$

Question 10

Suppose $$A is a$$ $$3 	imes 2$$ matrix. Which of the following statements about $$A^T A is $$true?

Accepted Answer

$$A^T A is a$$ $$2 	imes 2$$ symmetric matrix$

Question 11

If $$A is a$$ $$2 	imes 2$$ matrix with singular values $$\$$sigma_1$$ = 5$$ and $$\$$sigma_2$$ = 0$$, which of the following best describes $$A$$?

Accepted Answer

$$A is $$rank 1$

Question 12

Which of the following steps is NOT part of finding the SVD of a matrix $$A$$?

Accepted Answer

Multiplying $$A by $$its inverse to get $$U$$

Question 13

Let $$A be a$$ $$2 	imes 2$$ matrix with $$A^T A = \begin{pmatrix} 10 & 0 \ 0 & 5 \end{pmatrix}. $$What are the singular values of $$A$$?

Accepted Answer

$$\sqrt{10}$$ and $$\sqrt{5}$$

Question 14

If $$A is a$$ $$3 	imes 3$$ matrix and $$A^T A$$ has eigenvalues $$16$$, $$9$$, and $$0$$, what is the rank of $$A$$?

Accepted Answer

$$2$$

Question 15

Which of the following is true about the columns of $$U in $$the SVD $$A = U \Sigma V^T$$?

Accepted Answer

The columns of $$U$$ form an orthonormal basis for $$\mathbb{R}^m$$

Question 16

Suppose $$A is a$$ $$2 	imes 2$$ matrix with singular values $$\$$sigma_1$$ = 7$$ and $$\$$sigma_2$$ = 3. $$What is $$	ext{trace}(A^T A)$$?

Accepted Answer

$$58$$

Question 17

Which of the following is a property of the matrix $$\Sigma in $$the SVD $$A = U \Sigma V^T$$?

Accepted Answer

$$\Sigma is a $$diagonal matrix with non-negative real entries$

Question 18

If $$A is a$$ $$3 	imes 2$$ matrix and $$A^T A$$ has eigenvalues $$25$$ and $$4$$, what are the singular values of $$A$$?

Accepted Answer

$$5$$ and $$2$$

Question 19

Which of the following is NOT a use of singular value decomposition in data analysis?

Accepted Answer

Calculating the determinant of a matrix

Singular Value Decomposition (SVD) and Singular Values

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