Singular value decomposition (svd) in MATLAB - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is Singular Value Decomposition (SVD) in MATLAB?

SVD is a method to break down a matrix into three simpler matrices: U, S, and V, where U and V are orthogonal, and S is diagonal with singular values.

beginner

What does the MATLAB command [U,S,V] = svd(A) return?

It returns matrices U, S, and V such that A = U * S * V'. U and V are orthogonal matrices, and S is a diagonal matrix with singular values.

intermediate

Why are singular values in matrix S always non-negative?

Because singular values represent the lengths of the semi-axes of the ellipse formed by the matrix transformation, which cannot be negative.

intermediate

How can SVD help in image compression in MATLAB?

By keeping only the largest singular values and corresponding vectors, you can approximate the image with fewer data, reducing size while keeping important features.

intermediate

What is the shape of matrices U, S, and V when A is m-by-n?

U is m-by-m, S is m-by-n diagonal, and V is n-by-n. This ensures the multiplication U*S*V' reconstructs A.

In MATLAB, what does the matrix S contain after [U,S,V] = svd(A)?

A Random values

B Eigenvalues of A

C Inverse of A

D Singular values on the diagonal

Matrix S contains the singular values of A on its diagonal, sorted from largest to smallest.

Which MATLAB command reconstructs the original matrix A from its SVD components?

A U * S * V'

B U + S + V

C S * U * V

D V * S * U'

The original matrix A is reconstructed by multiplying U, S, and the transpose of V: A = U * S * V'.

What property do matrices U and V have in SVD?

A They are zero matrices

B They are diagonal matrices

C They are orthogonal matrices

D They are identity matrices

U and V are orthogonal matrices, meaning their columns are perpendicular unit vectors.

If A is 4-by-3, what is the size of matrix V in [U,S,V] = svd(A)?

A 3-by-4

B 3-by-3

C 4-by-3

D 4-by-4

Matrix V is n-by-n, so for A of size 4-by-3, V is 3-by-3.

Why might you use only the first few singular values in SVD?

A To approximate the matrix with less data

B To make the matrix larger

C To invert the matrix

D To find eigenvalues

Using only the largest singular values helps approximate the matrix while reducing data size, useful in compression.

Explain how Singular Value Decomposition breaks down a matrix and what each component represents.

Think of U and V as rotations and S as scaling.

Describe a practical use of SVD in MATLAB, such as image compression, and how it works.

Focus on how fewer singular values can still keep the main picture.

Practice

1. What does the MATLAB function svd(A) return when applied to a matrix A?

easy

A. The determinant of matrix A.

B. The inverse of matrix A.

C. Three matrices: U, S, and V such that A = U * S * V'.

D. The transpose of matrix A.

Solution

1. Step 1: Understand the purpose of svd

   The svd function decomposes a matrix into three parts: U, S, and V.

2. Step 2: Recall the decomposition formula

   It satisfies the equation A = U * S * V', where U and V are orthogonal matrices and S is diagonal.

3. Final Answer:

   Three matrices: U, S, and V such that A = U * S * V'. -> Option C

4. Quick Check:

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

Hint: Remember svd splits matrix into U, S, V parts [OK]

Common Mistakes:

• Confusing svd with inverse or transpose
• Expecting a single matrix output
• Thinking svd returns determinant

2. Which of the following is the correct MATLAB syntax to compute the singular value decomposition of matrix A and store the results in U, S, and V?

easy

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

B. svd(U, S, V) = A;

C. U, S, V = svd(A);

D. svd(A, U, S, V);

Solution

1. Step 1: Recall MATLAB function output syntax

   In MATLAB, multiple outputs are assigned using square brackets on the left side.

2. Step 2: Match correct svd syntax

   The correct syntax is [U, S, V] = svd(A); to get all three matrices.

3. Final Answer:

   [U, S, V] = svd(A); -> Option A

4. Quick Check:

   Use square brackets for multiple outputs [OK]

Hint: Use square brackets for multiple outputs in MATLAB [OK]

Common Mistakes:

• Using parentheses instead of brackets for outputs
• Trying to assign outputs on the right side
• Incorrect argument order in svd

3. Given the MATLAB code:

A = [3 1; 1 3];
[U, S, V] = svd(A);
disp(diag(S));

What will be the output of disp(diag(S))?

medium

A. [0; 0]

B. [3; 1]

C. [1; 3]

D. [4; 2]

Solution

1. Step 1: Calculate singular values of A

   Matrix A is symmetric. Its singular values are the absolute values of eigenvalues. Eigenvalues of A are 4 and 2.

2. Step 2: Understand svd output

   The diagonal of S contains singular values in descending order: 4 and 2.

3. Final Answer:

   [4; 2] -> Option D

4. Quick Check:

   Singular values = eigenvalues for symmetric matrix [OK]

Hint: For symmetric matrices, singular values = absolute eigenvalues [OK]

Common Mistakes:

• Confusing singular values with matrix entries
• Mixing order of singular values
• Expecting singular values to be zero

4. Consider the MATLAB code snippet:

A = [1 2; 3 4];
[U, S, V] = svd(A);
U = svd(A);

What is the error in this code?

medium

A. Matrix A is not square, so svd cannot be applied.

B. The second call to svd assigns a matrix to U instead of three outputs.

C. The variable V is not assigned in the second svd call.

D. The svd function requires four output variables.

Solution

1. Step 1: Analyze the first svd call

   The first call correctly assigns three outputs: U, S, V.

2. Step 2: Analyze the second svd call

   The second call assigns U = svd(A); which returns only singular values as a vector, not a matrix U.

3. Final Answer:

   The second call to svd assigns a matrix to U instead of three outputs. -> Option B

4. Quick Check:

   svd(A) alone returns singular values vector [OK]

Hint: svd(A) alone returns singular values vector, not matrices [OK]

Common Mistakes:

• Expecting svd(A) to return U matrix alone
• Thinking svd requires square matrix only
• Ignoring number of outputs in function call

5. You want to approximate a matrix A by keeping only the largest singular value and corresponding vectors using MATLAB. Which code snippet correctly performs this low-rank approximation?

hard

A. [U, S, V] = svd(A);A_approx = U(:,1) * S(1,1) * V(:,1)';

B. [U, S, V] = svd(A);A_approx = U(:,1) * V(:,1)';

C. [U, S, V] = svd(A);A_approx = S(1,1) * V(:,1)';

D. [U, S, V] = svd(A);A_approx = U(:,1) * S(:,1) * V(:,1)';

Solution

1. Step 1: Understand low-rank approximation using svd

   Keeping only the largest singular value means using the first column of U, the first singular value, and the first column of V.

2. Step 2: Construct the approximation

   The approximation is A_approx = U(:,1) * S(1,1) * V(:,1)', multiplying the first singular value and vectors.

3. Final Answer:

   [U, S, V] = svd(A); A_approx = U(:,1) * S(1,1) * V(:,1)'; -> Option A

4. Quick Check:

   Low-rank approx = U(:,1)*S(1,1)*V(:,1)' [OK]

Hint: Multiply first U column, singular value, and V column for approximation [OK]

Common Mistakes:

• Omitting singular value in multiplication
• Using entire S matrix instead of first singular value
• Incorrect matrix dimensions in multiplication