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

Question 1

Which of the following best describes the Singular Value Decomposition (SVD) of a 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

In the context of image processing, why is SVD particularly useful?

Accepted Answer

It allows for the compression of images by keeping only the largest singular values, reducing storage while preserving important features.

Question 3

Given a matrix $$A of $$size $$m 	imes n$$, what are the dimensions of the matrices $$U$$, $$\Sigma$$, and $$V in $$its SVD $$A = U \Sigma V^T$$?

Accepted Answer

$$U is$$ $$m 	imes m$$, $$\Sigma is$$ $$m 	imes n$$, $$V is$$ $$n 	imes n$$

Question 4

Which property is true for the singular values in $$\Sigma in $$the SVD of a real matrix $$A$$?

Accepted Answer

They are always non-negative and usually ordered from largest to smallest.

Question 5

Suppose you have a $$4 	imes 4$$ image matrix $$A$$ with rank 1. What does this imply about its SVD?

Accepted Answer

$$A$$ can be written as the outer product of two vectors, and only one singular value is nonzero.

Question 6

If $$A is a$$ $$3 	imes 2$$ matrix, what is the maximum possible number of nonzero singular values in its SVD?

Accepted Answer

2

Question 7

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

Accepted Answer

The columns of $$U$$ and $$V$$ are orthonormal vectors.

Question 8

How does SVD help in noise reduction for images?

Accepted Answer

By reconstructing the image using only the largest singular values, which capture the main features and ignore small, noisy components.

Question 9

Given a matrix $$A$$, which of the following is true about the eigenvalues of $$A^T A in $$relation to the singular values of $$A$$?

Accepted Answer

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

Question 10

A grayscale image is represented as a $$256 	imes 256$$ matrix. If you keep only the top 10 singular values in its SVD, what is the main effect?

Accepted Answer

The image is compressed, losing some detail but retaining the main structure.

Question 11

Which of the following is a real-world application of SVD outside of image processing?

Accepted Answer

Recommender systems, such as those used by streaming services to suggest movies.

Question 12

If $$A is a$$ $$5 	imes 3$$ matrix of rank 2, how many nonzero singular values does $$A$$ have?

Accepted Answer

2

Question 13

Which step is NOT part of computing the SVD of a matrix $$A$$?

Accepted Answer

Multiplying $$A by $$its inverse.

Question 14

Suppose $$A is a$$ $$3 	imes 3$$ matrix with SVD $$A = U \Sigma V^T. If$$ $$A is $$symmetric, which of the following is true?

Accepted Answer

$$U$$ and $$V$$ can be chosen to be the same orthogonal matrix.

Question 15

Which of the following best explains why SVD is used for low-rank approximation of matrices?

Accepted Answer

The sum of the first $$k$$ terms in the SVD gives the best rank-$$k$$ approximation in terms of the Frobenius norm.

Question 16

If $$A is a$$ $$2 	imes 2$$ matrix with singular values $$5$$ and $$2$$, what is the Frobenius norm of $$A$$?

Accepted Answer

$$\sqrt{5^2$ + 2^2$} = \sqrt{29}$$

Singular Value Decomposition (SVD) and Image Processing: Precalculus Study Notes

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