Understanding Eigenvalues and Eigenvectors

What Are Eigenvectors?

Eigenvectors are special directions in space that remain unchanged (except for scaling) when a linear transformation is applied. For a square matrix A, an eigenvector v satisfies:

$$Av = λv$$

where λ (lambda) is a scalar called the eigenvalue. It tells us how much the vector stretches or shrinks along that direction.

Why Do They Matter?

Eigenvectors reveal the fundamental directions of transformation in linear systems. They’re used in:

• Principal Component Analysis (PCA) for dimensionality reduction
• Google’s PageRank algorithm
• Image compression
• Quantum mechanics
• Vibration analysis in engineering

Finding Eigenvectors

Step 1: Find the Eigenvalues

Solve the characteristic equation:

$$\det(A - λI) = 0$$

where I is the identity matrix.

Step 2: Find the Eigenvectors

For each eigenvalue λ, solve:

$$(A - λI)x = 0$$

The non-zero solutions x are the eigenvectors.

Visual Example

When a transformation is applied, the dog stretches along the eigenvector direction

The image shows how an eigenvector maintains its direction under transformation, only changing in magnitude by the factor λ.