Summary  
This chapter demonstrates how to define a matrix in Python, compute its eigenvalues and eigenvectors using numpy’s eig function, and validate each eigenpair by checking that A @ v equals λ * v.  

General domain of usage  
Scientific computing

## Computing Eigenvalues and Eigenvectors

import numpy as np
from numpy.linalg import eig

# Define matrix A (square matrix)
A = np.array([[2, 1],
              [1, 2]])

# Solve for eigenvalues and eigenvectors
eigenvalues, eigenvectors = eig(A)

# Print eigenvalues and eigenvectors
print(f'Eigenvalues:\n{eigenvalues}')
print(f'Eigenvectors:\n{eigenvectors}')

`eig()` from the `numpy` library computes the solutions to the equation:

$$
A v = \lambda v
$$

* `eigenvalues`: a list of scalars $$\lambda$$ that scale eigenvectors;
* `eigenvectors`: columns representing $$v$$ (directions that don't change under transformation).

import numpy as np
from numpy.linalg import eig

# Define matrix A (square matrix)
A = np.array([[2, 1],
              [1, 2]])

# Solve for eigenvalues and eigenvectors
eigenvalues, eigenvectors = eig(A)

# Verify that A @ v = λ * v for each eigenpair
for i in range(len(eigenvalues)):
    print(f'Pair {i + 1}:')
    λ = eigenvalues[i]
    v = eigenvectors[:, i].reshape(-1, 1)
    print(f'A * v:\n{A @ v}')
    print(f'lambda * v:\n{λ * v}')

This checks if:

$$
A v = \lambda v
$$

The two sides should match closely, which confirms correctness. This is how we validate theoretical properties numerically.

Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

Master the concepts of sets and series, from basic operations to practical applications. Gain hands-on experience implementing set operations and working with arithmetic and geometric series in Python.

Develop a solid understanding of limits, derivatives, integrals, and partial derivatives. Connect theory to practice by implementing these concepts in Python and applying them to optimization through gradient descent.

Build strong knowledge of vectors, matrices, and transformations. Learn decomposition methods and eigenvalue analysis, while reinforcing concepts with Python coding challenges and practical data science applications.

Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Implementing Eigenvectors & Eigenvalues in Python

Computing Eigenvalues and Eigenvectors

Validating Each Pair (Key Step)