Singular value decomposition (svd) in MATLAB - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to perform singular value decomposition on matrix A.

MATLAB

[U, S, V] = [1](A);

Drag options to blanks, or click blank then click option'

Aeig

Bdet

Cinv

Dsvd

Attempts:

3 left

💡 Hint

Common Mistakes

Using eigenvalue function eig instead of svd.

Trying to invert the matrix with inv.

The svd function in MATLAB computes the singular value decomposition of a matrix.

2fill in blank

medium

Complete the code to extract the singular values from matrix A.

MATLAB

S = [1](A);

Drag options to blanks, or click blank then click option'

Arank

Beig

Csvd

Ddet

Attempts:

3 left

💡 Hint

Common Mistakes

Using eig which returns eigenvalues, not singular values.

Using rank or det which do not return singular values.

The svd function returns singular values when called with one output.

3fill in blank

hard

Fix the error in the code to compute the SVD of matrix B.

MATLAB

[U, S, V] = svd([1]);

Drag options to blanks, or click blank then click option'

Ab

BB

CB()

DB[]

Attempts:

3 left

💡 Hint

Common Mistakes

Using lowercase b instead of B.

Adding parentheses or brackets after the variable name.

Variable names in MATLAB are case-sensitive. Use B exactly as defined.

4fill in blank

hard

Fill both blanks to create a diagonal matrix of singular values and reconstruct matrix A.

MATLAB

S = diag([1]);
A_reconstructed = [2] * S * V';

Drag options to blanks, or click blank then click option'

As

BU

CV

DS

Attempts:

3 left

💡 Hint

Common Mistakes

Using S instead of the vector s for diag.

Using V instead of U for reconstruction.

The vector of singular values is s. The matrix U is used to reconstruct A with S and V.

5fill in blank

hard

Fill all three blanks to compute the reduced SVD of matrix M.

MATLAB

[[1], [2], [3]] = svd(M, 'econ');

Drag options to blanks, or click blank then click option'

AU

BS

CV

DE

Attempts:

3 left

💡 Hint

Common Mistakes

Using E which is not a standard output of svd.

Mixing the order of U, S, and V.

The reduced SVD returns U, S, and V matrices.

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