Eigenvalues and eigenvectors (eig) in MATLAB - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of eigenvalues and eigenvectors for a 2x2 matrix

What is the output of the following MATLAB code?

A = [2 1; 1 2]; [V,D] = eig(A); disp(D); disp(V);

MATLAB

A = [2 1; 1 2];
[V,D] = eig(A);
disp(D);
disp(V);

A

[1 0; 0 3]
[[-0.7071 0.7071]; [0.7071 0.7071]]

B

[3 0; 0 1]
[[0.7071 -0.7071]; [0.7071 0.7071]]

C

[2 1; 1 2]
[[1 0]; [0 1]]

D

[0 0; 0 0]
[[0 0]; [0 0]]

💡 Hint

Remember that eig returns eigenvalues in ascending order on the diagonal of D and corresponding eigenvectors in V.

The matrix A has eigenvalues 1 and 3. MATLAB's eig returns them in ascending order on the diagonal of D. The columns of V are the corresponding eigenvectors normalized to length 1.

🧠 Conceptual

intermediate

Understanding eigenvalue multiplicity

Consider the matrix A = [2 0; 0 2]. How many distinct eigenvalues does A have?

A Two distinct eigenvalues

B One eigenvalue with multiplicity two

C No eigenvalues

D Infinite eigenvalues

💡 Hint

Look at the diagonal elements and their values.

Matrix A is diagonal with both diagonal entries equal to 2. This means it has one eigenvalue 2 with algebraic multiplicity 2.

🔧 Debug

advanced

Identify the error in eigenvalue calculation code

What error will the following MATLAB code produce?

A = [1 2; 3 4]; [V,D] = eig(A(1:3,1:3));

MATLAB

A = [1 2; 3 4];
[V,D] = eig(A(1:3,1:3));

A Undefined function or variable 'eig'.

B No error, outputs eigenvalues and eigenvectors.

C Matrix must be symmetric.

D Index exceeds matrix dimensions.

💡 Hint

Check the size of matrix A and the indexing used.

Matrix A is 2x2, but the code tries to access rows and columns 1 to 3, which is out of bounds, causing an indexing error.

📝 Syntax

advanced

Syntax error in eigenvalue function usage

Which option contains a syntax error when computing eigenvalues and eigenvectors in MATLAB?

A V, D = eig(A);

B [V,D] = eig A;

C [V,D] = eig(A, 'nobalance');

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

💡 Hint

MATLAB uses square brackets for multiple outputs and does not support comma-separated assignment.

Option A uses Python-style multiple assignment with commas, which is invalid in MATLAB syntax.

🚀 Application

expert

Number of eigenvalues for a 3x3 matrix with repeated eigenvalues

Given the matrix B = [5 4 2; 0 1 0; 0 0 1], how many distinct eigenvalues does B have?

A One eigenvalue with multiplicity three

B No eigenvalues

C Two distinct eigenvalues

D Three distinct eigenvalues

💡 Hint

Look for eigenvalues on the diagonal and consider repeated values.

Matrix B is upper triangular, so eigenvalues are diagonal entries: 5, 1, 1. So two distinct eigenvalues: 5 and 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