Linear Algebra: Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra that help us understand how certain transformations affect vectors in space.

Core Concept

When a square matrix A acts on a vector v, it typically changes both the direction and magnitude of the vector. However, there are special vectors that, when transformed by matrix A, only get scaled (stretched or compressed) without changing direction. These special vectors are called eigenvectors, and the scaling factors are called eigenvalues.

Finding Eigenvalues and Eigenvectors

To find the eigenvalues and eigenvectors of a matrix A:

1. Start with the eigenvalue equation: Av = λv
2. Rearrange to: (A - λI)v = 0, where I is the identity matrix
3. For non-trivial solutions (where v ≠ 0), we need: det(A - λI) = 0
4. This equation is called the characteristic equation, and its solutions are the eigenvalues
5. For each eigenvalue λ, solve (A - λI)v = 0 to find the corresponding eigenvectors

Properties of Eigenvalues and Eigenvectors

• A matrix of size n×n can have at most n distinct eigenvalues
• Eigenvectors corresponding to different eigenvalues are linearly independent
• The determinant of a matrix equals the product of its eigenvalues
• The trace (sum of diagonal elements) equals the sum of eigenvalues
• If λ is an eigenvalue of A, then λ^k is an eigenvalue of A^k

Diagonalization