Eigenvalues and eigenvectors (eig) in MATLAB

Introduction

Eigenvalues and eigenvectors help us understand important properties of a matrix, like how it stretches or rotates space.

To find the main directions in data for dimensionality reduction.

To analyze stability in systems like engineering or physics.

To solve systems of linear differential equations.

To understand vibrations or modes in mechanical structures.

Syntax

MATLAB

[V,D] = eig(A)

A is a square matrix.

V is a matrix whose columns are eigenvectors of A.

D is a diagonal matrix with eigenvalues of A on the diagonal.

Examples

Find eigenvalues and eigenvectors of a 2x2 matrix.

MATLAB

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

Eigenvalues are the diagonal elements, eigenvectors are standard basis vectors.

MATLAB

A = [4 0 0; 0 3 0; 0 0 2];
[V,D] = eig(A);

Matrix with complex eigenvalues and eigenvectors.

MATLAB

A = [0 -1; 1 0];
[V,D] = eig(A);

Sample Program

This program calculates and shows eigenvalues and eigenvectors of matrix A.

MATLAB

A = [3 1; 0 2];
[V,D] = eig(A);
disp('Eigenvalues:');
disp(diag(D));
disp('Eigenvectors:');
disp(V);

Important Notes

Eigenvalues can be real or complex numbers.

Eigenvectors are usually normalized (length 1) in MATLAB output.

Order of eigenvalues in D matches columns of V.

Summary

Use eig to find eigenvalues and eigenvectors of square matrices.

Eigenvalues tell how the matrix scales vectors; eigenvectors show directions.

Useful in many fields like data analysis, physics, and engineering.

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