Eigenvectors are a fundamental concept in linear algebra, with applications spanning across various fields such as physics, computer science, and data science. In this blog post, we’ll explore what eigenvectors are, why they are important, and how they are used in real-world scenarios.

What is an Eigenvector?

In simple terms, an eigenvector of a matrix is a non-zero vector that changes at most by a scalar factor when that matrix is applied to it. Mathematically, for a given matrix A, an eigenvector v satisfies the equation:

$$A \mathbf{v} = \lambda \mathbf{v}$$

where $λ$ (lambda) is the eigenvalue corresponding to the eigenvector v.

Why are Eigenvectors Important?

Eigenvectors and eigenvalues have several important properties and applications:

1. Principal Component Analysis (PCA): In data science, PCA is a technique used to reduce the dimensionality of data while preserving as much variance as possible. Eigenvectors of the covariance matrix of the data are used to determine the principal components.

2. Stability Analysis: In systems theory, eigenvectors are used to analyze the stability of equilibrium points in dynamic systems.

3. Quantum Mechanics: In quantum mechanics, eigenvectors and eigenvalues are used to describe the states of a quantum system and their corresponding energies.

Principal Component Analysis

Let’s see how eigenvectors are used in PCA with a simple example. Suppose we have a dataset with two features, and we want to reduce it to one feature while preserving as much variance as possible.

First, we compute the covariance matrix of the data. Then, we find the eigenvectors and eigenvalues of this covariance matrix. The eigenvector corresponding to the largest eigenvalue gives us the direction of the maximum variance.

Interactive Eigenvector Example

To make things more interactive, let’s visualize how a matrix transforms an eigenvector. Adjust the matrix values and see how the eigenvector changes.