Singular value decomposition (svd) in MATLAB - Step-by-Step Execution

Concept Flow - Singular value decomposition (svd)

Start with matrix A

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Compute U, S, V such that A = U*S*V'

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U: orthogonal matrix (left singular vectors)

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S: diagonal matrix (singular values)

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V: orthogonal matrix (right singular vectors)

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Use U, S, V for analysis or reconstruction

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End

Start with matrix A, compute matrices U, S, V so that A equals U times S times V transpose, then use these matrices for further tasks.

Execution Sample

MATLAB

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

This code computes the singular value decomposition of matrix A and shows the matrices U, S, and V.

Execution Table

Step	Action	Evaluation	Result
1	Define matrix A	A = [3 1 1; -1 3 1]	A = [3 1 1; -1 3 1]
2	Call svd(A)	Compute U, S, V such that A = U*S*V'	U, S, V matrices computed
3	Display U	Show left singular vectors	U = [0.71 0.71; 0.71 -0.71] (approx)
4	Display S	Show singular values on diagonal	S = diag([3.87, 2.24]) (approx)
5	Display V	Show right singular vectors	V = [0.82 0.41 0.41; -0.45 0.89 0; 0.37 0.18 -0.91] (approx)
6	Verify reconstruction	Calculate U*S*V'	Reconstructed A matches original A
7	End	No more steps	Execution complete

💡 All matrices computed and verified, svd process complete

Variable Tracker

Variable	Start	After svd	Final
A	[3 1 1; -1 3 1]	[3 1 1; -1 3 1]	[3 1 1; -1 3 1]
U	undefined	[0.71 0.71; 0.71 -0.71]	[0.71 0.71; 0.71 -0.71]
S	undefined	[3.87 0 0; 0 2.24 0]	[3.87 0 0; 0 2.24 0]
V	undefined	[0.82 0.41 0.41; -0.45 0.89 0; 0.37 0.18 -0.91]	[0.82 0.41 0.41; -0.45 0.89 0; 0.37 0.18 -0.91]

Key Moments - 3 Insights

Why is matrix S diagonal and not full?

Matrix S contains singular values only on its diagonal, representing the strength of each component. This is shown in execution_table step 4 where S is diagonal with singular values.

Why do we multiply U, S, and V' to get back A?

Because svd decomposes A into U, S, and V such that A = U*S*V'. This is verified in execution_table step 6 where reconstruction matches original A.

Are U and V always square and orthogonal?

Yes, U and V are orthogonal matrices with orthonormal columns, as shown in execution_table steps 3 and 5.

Visual Quiz - 3 Questions

Look at the execution_table at step 4, what does matrix S represent?

A Matrix of left singular vectors

B Matrix of right singular vectors

C Diagonal matrix of singular values

D Original matrix A

💡 Hint

Refer to execution_table row 4 where S is described as singular values on diagonal

At which step in the execution_table do we verify that U*S*V' reconstructs A?

A Step 3

B Step 6

C Step 5

D Step 2

💡 Hint

Check execution_table row 6 where reconstruction is verified

If matrix A was square and symmetric, what would you expect about U and V?

A U and V would be identical

B U would be zero matrix

C V would be diagonal

D S would be zero matrix

💡 Hint

Recall that for symmetric matrices, U and V are equal (related to eigen decomposition)

Concept Snapshot

Singular value decomposition (svd) splits matrix A into U, S, V such that A = U*S*V'.
U and V are orthogonal matrices with singular vectors.
S is diagonal with singular values.
Use svd to analyze or reconstruct A.
In MATLAB: [U,S,V] = svd(A);

Full Transcript

Singular value decomposition (svd) breaks a matrix A into three matrices: U, S, and V. U and V are orthogonal matrices containing singular vectors, and S is a diagonal matrix with singular values. The product U*S*V' reconstructs the original matrix A. In MATLAB, you use [U,S,V] = svd(A) to compute these matrices. This process helps analyze the structure of A and is useful in many applications like data compression and noise reduction.

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