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Chapter 6: Eigenvalues and Eigenvectors - A Comprehensive Guide

Eigenvalues and eigenvectors are fundamental concepts in linear algebra with applications spanning diverse fields like physics, engineering, data science, and computer graphics. They reveal inherent properties of linear transformations represented by matrices, offering insights into stability, oscillations, dimensionality reduction, and much more. This tutorial provides a comprehensive understanding of eigenvalues and eigenvectors, covering theoretical foundations, practical computation, and illustrative code examples in Python using NumPy.

*1. Introduction: What are Eigenvalues and Eigenvectors?*

At their core, eigenvalues and eigenvectors describe special vectors that remain unchanged in direction when a linear transformation is applied. They are scaled by a factor called the eigenvalue. Imagine a stretching or rotation transformation. Eigenvectors are those vectors which, after the transformation, still point in the same direction, just potentially longer or shorter.

*Formal Definition:*

Let A be an n x n matrix. A non-zero vector *v* is an eigenvector of A if there exists a scalar λ (lambda), called an *eigenvalue*, such that:

*A v = λ v*

*A:* The n x n square matrix representing the linear transformation.
*v:* The eigenvector (a non-zero vector).
*λ:* The eigenvalue (a scalar).

*In simpler terms:* Applying the matrix A to the eigenvector *v* simply scales *v* by the factor λ.

*Geometric Interpretation:*

*Eigenvector:* Defines a direction that is invariant (up to scaling) under the linear transformation A.
*Eigenvalue:* Represents the factor by which the eigenvector is scaled when the transformation A is applied.

*Example:*

Consider a matrix A that stretches vectors in the x-direction by a factor of 2 and leaves the y-direction unchanged. The vector (1, 0) would be an eigenvector with eigenvalue 2, because A * (1, 0) = 2 * (1, 0) ...

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