gen-define-eigenvalues-and-eigenvectors-and-give-one-short-worke-c568

Define eigenvalues and eigenvectors and give one short worked example.

# Eigenvalues and Eigenvectors ## Definitions For a square matrix **A**, a scalar λ (lambda) is an **eigenvalue** and a non-zero vector **v** is an **eigenvector** if: **A**/**v** = λ**v** In words: When matrix **A** transforms vector **v**, the result is just a scaled version of **v** (scaled by λ). ## Worked Example Find the eigenvalues and eigenvectors of **A** = [4 1] [2 3] **Step 1:** Find eigenvalues by solving det(**A** - λ**I**) = 0 **A** - λ**I** = [4-λ 1 ] [2 3-λ ] det(**A** - λ**I**) = (4-λ)(3-λ) - (1)(2) = λ² - 7λ + 10 = 0 (λ - 5)(λ - 2) = 0 **Eigenvalues: λ₁ = 5, λ₂ = 2** **Step 2:** Find eigenvector for λ₁ = 5 (**A** - 5**I**)**v** = **0** → [-1 1][v₁] = [0] [2 -2][v₂] [0] This gives: -v₁ + v₂ = 0, so v₂ = v₁ **Eigenvector: v₁ = [1]** (or any scalar multiple) [1] **Step 3:** Find eigenvector for λ₂ = 2 (**A** - 2**I**)**v** = **0** → [2 1][v₁] = [0] [2 1][v₂] [0] This gives: 2v₁ + v₂ = 0, so v₂ = -2v₁ **Eigenvector: v₂ = [ 1]** (or any scalar multiple) [-2]