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.