Singular value decomposition (svd) in MATLAB - Mini Project: Build & Apply

Singular value decomposition (svd)

📖 Scenario: You work as a data analyst and want to understand the structure of a matrix representing some data. Singular value decomposition (SVD) helps break down the matrix into simpler parts.

🎯 Goal: Learn how to perform singular value decomposition on a matrix in MATLAB and display the decomposed matrices.

📋 What You'll Learn

Create a matrix variable named A with exact values

Create a variable k to select the number of singular values

Use MATLAB's svd function to decompose the matrix A

Display the matrices U, S, and V from the decomposition

💡 Why This Matters

🌍 Real World

Singular value decomposition is used in image compression, noise reduction, and data analysis to simplify complex data.

💼 Career

Understanding SVD is important for data scientists, engineers, and researchers working with large datasets and machine learning.

1

Create the matrix A

Create a 3x3 matrix called A with these exact values: first row [1 2 3], second row [4 5 6], third row [7 8 9].

MATLAB

% Create the matrix A with the specified values
% Your code here

Hint

Use square brackets and semicolons to create rows in MATLAB.

2

Set the number of singular values k

Create a variable called k and set it to 2 to select the first two singular values.

MATLAB

A = [1 2 3; 4 5 6; 7 8 9];
% Create the variable k and set it to 2
% Your code here

Hint

Just assign the number 2 to the variable k.

3

Perform singular value decomposition

Use the MATLAB svd function to decompose matrix A into U, S, and V. Write [U, S, V] = svd(A);.

MATLAB

A = [1 2 3; 4 5 6; 7 8 9];
k = 2;
% Perform SVD on A to get U, S, and V
% Your code here

Hint

Use square brackets to capture multiple outputs from svd.

4

Display the matrices U, S, and V

Use disp to print the matrices U, S, and V in order.

MATLAB

A = [1 2 3; 4 5 6; 7 8 9];
k = 2;
[U, S, V] = svd(A);
% Display U, S, and V matrices
% Your code here

Hint

Use disp('Matrix U:') before displaying U to label the output.

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