Summary  
The chapter explains how to compute eigenvalues and eigenvectors of a square matrix using numerical methods, demonstrating both the direct approach with numpy.linalg.eig and the power iteration algorithm for estimating the dominant eigenpair.

General domain of usage  
Dimensionality reduction (e.g., Principal Component Analysis)

**Video: Introduction to Eigenvalues and Eigenvectors**

This video explains what **eigenvalues** and **eigenvectors** are, how they are defined mathematically, and highlights their significance and applications in data science, including their use in techniques like **Principal Component Analysis (PCA)**.

Eigenvalues and eigenvectors are central concepts in linear algebra that help you understand how linear transformations affect data. Mathematically, for a given square matrix $$A$$, an eigenvector $$v$$ is a nonzero vector that, when multiplied by $$A$$, results in a scalar multiple of itself: $$A v = λ v$$. Here, $$λ$$ is the eigenvalue corresponding to the eigenvector $$v$$. This means that applying the transformation $$A$$ to $$v$$ only stretches or compresses the vector, without changing its direction.

In data science, **eigenvalues** and **eigenvectors** are especially important in dimensionality reduction techniques like **Principal Component Analysis (PCA)**. PCA uses the eigenvectors of a data covariance matrix to identify directions (**principal components**) along which the data varies the most. The corresponding eigenvalues tell you how much variance is captured by each principal component, allowing you to reduce the dimensionality of your data while preserving its essential structure.

import numpy as np

# Define a 2x2 matrix
A = np.array([[4, 2],
              [1, 3]])

# Start with a random vector
v = np.array([1, 1], dtype=float)
v = v / np.linalg.norm(v)  # Normalize the vector

# Power iteration to estimate the dominant eigenvalue and eigenvector
for _ in range(10):
    v = A @ v
    v = v / np.linalg.norm(v)

# Estimate the dominant eigenvalue
eigenvalue = (A @ v).dot(v) / v.dot(v)

print("Estimated dominant eigenvalue:", eigenvalue)
print("Estimated dominant eigenvector:", v)

The code above uses the **power iteration method** to estimate the dominant eigenvalue and eigenvector of a 2x2 matrix. This iterative approach repeatedly multiplies a starting vector by the matrix and normalizes it, gradually aligning the vector with the direction of the dominant eigenvector. The dominant eigenvalue is then estimated by projecting the matrix-vector product onto the current eigenvector estimate.

While this method is simple and effective for finding the largest eigenvalue and its corresponding eigenvector, it has limitations. It may not converge quickly or accurately for matrices with closely spaced or negative eigenvalues, and it only finds the dominant pair. In practice, more robust algorithms—such as those implemented in libraries like `numpy.linalg.eig` – are used to compute all eigenvalues and eigenvectors efficiently.

Eigenvalues and eigenvectors are used in practice to analyze data structure, reduce dimensionality, and reveal patterns. For example, in **PCA**, you compute the eigenvectors of the covariance matrix to find new axes that capture the most variance, allowing you to project high-dimensional data onto a lower-dimensional space without losing critical information.

Which statement best describes the relationship between eigenvalues, eigenvectors, and their application in data science as discussed in this chapter?

Establish core mathematical intuition for data science by manipulating vectors, multiplying matrices, solving linear systems, and computing eigenvalues for dimensionality reduction, all utilizing Python.

All core linear algebra concepts for data science in one section, with each chapter building on the previous to develop a strong mathematical foundation using Python.