$$U is$$ $$3 \times 3$$, $$\Sigma is$$ $$3 \times 2$$, $$V is$$ $$2 \times 2$$

Question 1

Which of the following best describes the Singular Value Decomposition (SVD) of a real $$m 	imes n$$ 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 entries.

Question 2

Given a $$3 	imes 2$$ real matrix $$A$$, what are the dimensions of $$U$$, $$\Sigma$$, and $$V in $$the SVD $$A = U \Sigma V^T$$?

Accepted Answer

$$U is$$ $$3 	imes 3$$, $$\Sigma is$$ $$3 	imes 2$$, $$V is$$ $$2 	imes 2$$

Question 3

Which property do the singular values of a matrix $$A$$ always satisfy?

Accepted Answer

They are always non-negative real numbers.

Question 4

Suppose $$A is a$$ $$2 	imes 2$$ real matrix. Which of the following is a correct step in constructing the SVD of $$A$$?

Accepted Answer

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

Question 5

If $$A is a$$ $$3 	imes 2$$ matrix with rank 2, how many nonzero singular values does $$A$$ have?

Accepted Answer

2

Question 6

Which of the following statements about the matrices $$U$$ and $$V in $$the SVD $$A = U \Sigma V^T is $$correct?

Accepted Answer

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

Question 7

In the context of SVD, what is the relationship between the nonzero singular values of $$A$$ and the eigenvalues of $$A^T A$$?

Accepted Answer

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

Question 8

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

Accepted Answer

Image compression

Question 9

Given $$A = \begin{pmatrix} 1 & 0 \ 0 & 2 \end{pmatrix}$$, what are the singular values of $$A$$?

Accepted Answer

1 and 2

Question 10

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

Accepted Answer

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

Question 11

Which step is NOT part of the standard procedure for constructing the SVD of a matrix $$A$$?

Accepted Answer

Find the eigenvalues of $$A + A^T.$$

Question 12

If $$A is a$$ $$3 	imes 3$$ real matrix and $$	ext{rank}(A) = 1$$, how many nonzero singular values does $$A$$ have?

Accepted Answer

1

Question 13

Which of the following MATLAB commands would you use to compute the SVD of a matrix $$A$$?

Accepted Answer

[U, S, V] = svd(A)

Question 14

Why is SVD particularly useful for analyzing data matrices in applications such as Principal Component Analysis (PCA)?

Accepted Answer

It provides orthogonal directions that maximize variance.

Question 15

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

Accepted Answer

3 and 2

Question 16

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

Accepted Answer

They are the singular values of $$A$$ arranged in non-increasing order.

Singular Value Decomposition (SVD): Concepts, Construction, and Applications

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