Eigenvalues and eigenvectors (eig) in MATLAB - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to compute eigenvalues of matrix A.

MATLAB

A = [2 1; 1 2];
lambda = [1](A);

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

Adet

Brank

Cinv

Deig

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'inv' instead of 'eig' to compute eigenvalues.

Using 'det' which returns determinant, not eigenvalues.

The eig function computes eigenvalues of a matrix.

2fill in blank

medium

Complete the code to compute eigenvectors of matrix B.

MATLAB

B = [4 0; 1 3];
[V, D] = [1](B);

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

Aeig

Binv

Csvd

Dchol

Attempts:

3 left

💡 Hint

Common Mistakes

Using 'svd' which computes singular value decomposition, not eigenvectors.

Using 'chol' which computes Cholesky decomposition.

The eig function returns eigenvectors and eigenvalues when called with two outputs.

3fill in blank

hard

Fix the error in the code to correctly compute eigenvalues of matrix C.

MATLAB

C = [1 2; 3 4];
lambda = eig[1]C);

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A(

B[

C{

D<

Attempts:

3 left

💡 Hint

Common Mistakes

Using square brackets or curly braces instead of parentheses.

Missing the opening parenthesis causes syntax error.

The function call syntax requires parentheses around the argument: eig(C).

4fill in blank

hard

Fill both blanks to create a dictionary of eigenvalues and eigenvectors for matrix D.

MATLAB

D = [5 2; 2 5];
[V, [1]] = eig(D);
eigenvalues = diag([2]);

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

AD

BV

DV'

Attempts:

3 left

💡 Hint

Common Mistakes

Assigning eigenvectors to the second output instead of eigenvalues.

Using the wrong variable inside diag function.

The second output of eig is the diagonal matrix of eigenvalues, so D is assigned to {{BLANK_1}}. To get eigenvalues as a vector, use diag(D).

5fill in blank

hard

Fill all three blanks to compute eigenvalues and eigenvectors, then normalize eigenvectors of matrix E.

MATLAB

E = [3 1; 0 2];
[[1], [2]] = eig(E);
norm_vec = [3]( [1](:,1) );

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

AV

BD

Cnorm

Deig

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping eigenvectors and eigenvalues in assignment.

Using eig function instead of norm to normalize.

eig returns eigenvectors and eigenvalues. We assign eigenvectors to V and eigenvalues to D. To normalize the first eigenvector, use norm(V(:,1)).

Practice

1. What does the MATLAB function eig return when applied to a square matrix?

easy

A. Eigenvalues and eigenvectors of the matrix

B. Only the eigenvalues of the matrix

C. Only the eigenvectors of the matrix

D. The inverse of the matrix

Solution

1. Step 1: Understand the purpose of eig

   The eig function in MATLAB is designed to compute eigenvalues and eigenvectors of a square matrix.

2. Step 2: Recall the output of eig

   When called with two output arguments, [V,D] = eig(A), it returns eigenvectors in V and eigenvalues in diagonal matrix D.

3. Final Answer:

   Eigenvalues and eigenvectors of the matrix -> Option A

4. Quick Check:

   eig returns eigenvalues and eigenvectors [OK]

Hint: Remember: eig returns both eigenvalues and eigenvectors [OK]

Common Mistakes:

• Thinking eig returns only eigenvalues
• Confusing eig with inverse function
• Assuming eig returns a scalar

2. Which of the following is the correct syntax to get eigenvalues and eigenvectors of matrix A in MATLAB?

easy

A. [V,D] = eig(A);

B. eig = [V,D](A);

C. V, D = eig(A);

D. eig(A) -> [V,D];

Solution

1. Step 1: Recall MATLAB function call syntax

   In MATLAB, functions return outputs using square brackets for multiple outputs, e.g., [out1, out2] = function(input);.

2. Step 2: Match correct syntax for eig

   The correct way to get eigenvectors V and eigenvalues D is [V,D] = eig(A);.

3. Final Answer:

   [V,D] = eig(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 commas instead of brackets for outputs
• Assigning outputs incorrectly
• Using arrow notation which is invalid in MATLAB

3. Given the matrix A = [2 0; 0 3];, what is the output of [V,D] = eig(A); D?

medium

A. [[5 0]; [0 0]]

B. [[3 0]; [0 2]]

C. [[2 0]; [0 3]]

D. [[0 2]; [3 0]]

Solution

1. Step 1: Identify eigenvalues of matrix A

   Matrix A is diagonal with entries 2 and 3, so eigenvalues are 2 and 3.

2. Step 2: Understand output of D from eig

   D is a diagonal matrix with eigenvalues on the diagonal, so D = [2 0; 0 3].

3. Final Answer:

   [[2 0]; [0 3]] -> Option C

4. Quick Check:

   Diagonal matrix D has eigenvalues 2 and 3 [OK]

Hint: Eigenvalues of diagonal matrix are diagonal entries [OK]

Common Mistakes:

• Mixing order of eigenvalues
• Confusing eigenvalues with eigenvectors
• Expecting eigenvalues as a vector instead of diagonal matrix

4. What is wrong with this MATLAB code snippet to get eigenvalues of matrix A?

eigenvalues = eig(A, 'vector');

medium

A. The matrix A must be square and symmetric

B. The variable name eigenvalues is invalid

C. The function eig requires three output arguments

D. The eig function does not accept 'vector' as a second argument

Solution

1. Step 1: Check eig function arguments

   The MATLAB eig function does not take a second argument like 'vector'. It only takes the matrix as input.

2. Step 2: Identify correct usage to get eigenvalues as vector

   To get eigenvalues as a vector, simply call eig(A) without extra arguments.

3. Final Answer:

   The eig function does not accept 'vector' as a second argument -> Option D

4. Quick Check:

   Check function syntax carefully [OK]

Hint: Check MATLAB function syntax in docs before adding arguments [OK]

Common Mistakes:

• Adding unsupported arguments to eig
• Assuming eig returns vector by default without output assignment
• Confusing eig with other functions

5. You have a matrix A = [4 1 0; 1 4 1; 0 1 4];. How can you find the eigenvector corresponding to the largest eigenvalue using MATLAB?

hard

A. largestVec = eig(A);

B. [V,D] = eig(A); [~, idx] = max(diag(D)); largestVec = V(:, idx);

C. [V,D] = eig(A); largestVec = D(:,1);

D. [V,D] = eig(A); largestVec = max(V);

Solution

1. Step 1: Compute eigenvalues and eigenvectors

   Use [V,D] = eig(A); to get eigenvectors V and eigenvalues in diagonal matrix D.

2. Step 2: Find index of largest eigenvalue

   Extract eigenvalues with diag(D), then find index of max eigenvalue with [~, idx] = max(diag(D));.

3. Step 3: Select eigenvector corresponding to largest eigenvalue

   Use index to get eigenvector: largestVec = V(:, idx);.

4. Final Answer:

   [V,D] = eig(A); [~, idx] = max(diag(D)); largestVec = V(:, idx); -> Option B

5. Quick Check:

   Index max eigenvalue, select eigenvector column [OK]

Hint: Use max(diag(D)) to find largest eigenvalue index [OK]

Common Mistakes:

• Using eigenvalues matrix D as vector directly
• Selecting wrong column of eigenvectors
• Confusing eigenvalues with eigenvectors