Chapter 6: Eigenvalues and Eigenvectors

This post is based on Introduction to Linear Algebra by Gilbert Strang book.

1. Definition of Eigenvalues and Eigenvectors

For a square matrix A, an eigenvector v is a non-zero vector that satisfies:

where λ is the corresponding eigenvalue. This equation means that multiplying A by v scales v by λ without changing its direction.

2. How to Find Eigenvalues and Eigenvectors

To find eigenvalues and eigenvectors:

1. Solve the characteristic equation:

where I is the identity matrix.

2. For each eigenvalue λ, solve the system:

to find the corresponding eigenvector v.

Example 1: 2x2 Matrix

Let’s compute the eigenvalues and eigenvectors of:

Step 1: Find Eigenvalues

Solve the characteristic equation:

Solve the quadratic equation:

Step 2: Find Eigenvectors

For λ1​=5:

This simplifies to:

So, the eigenvector corresponding to λ1​=5 is:

For λ2​=2:

This simplifies to:

So, the eigenvector corresponding to λ2​=2 is:

Python Code:

import numpy as np

# Define the matrix
A = np.array([[4, 1],
              [2, 3]])

# Compute eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(A)

print("Eigenvalues:")
print(eigenvalues)
print("\nEigenvectors:")
print(eigenvectors)

Note: The eigenvectors are normalized to unit length.

3. Applications of Eigenvalues and Eigenvectors

• Diagonalization: Expressing a matrix A as A=PDP−1, where D is a diagonal matrix of eigenvalues and P is a matrix of eigenvectors.
• Principal Component Analysis (PCA): A dimensionality reduction technique in machine learning that uses eigenvectors of the covariance matrix.
• Stability Analysis: In differential equations, eigenvalues determine the stability of a system.

Happy learning :-)