Linear Transformations
Eigenvalues and Eigenvectors
Eigenvalue equation Ax=λx, characteristic polynomial, finding eigenvalues and eigenvectors, diagonalization, geometric and algebraic multiplicity, applications.
Eigenvalues and Eigenvectors
Definition: For a square matrix A, a nonzero vector x is an eigenvector with eigenvalue λ if:
Ax = λx
i.e., A only scales x by factor λ (doesn't change direction)
Ax = λx
i.e., A only scales x by factor λ (doesn't change direction)
Finding Eigenvalues — Characteristic Polynomial
Ax = λx → (A−λI)x = 0
Non-trivial solution exists iff det(A−λI) = 0
This is the characteristic equation.
Non-trivial solution exists iff det(A−λI) = 0
This is the characteristic equation.
Example: Find eigenvalues of A = [[3,1],[0,2]]
det(A−λI) = det([[3−λ,1],[0,2−λ]]) = (3−λ)(2−λ) − 0 = 0
→ (3−λ)(2−λ) = 0 → λ₁ = 3, λ₂ = 2
Finding Eigenvectors
For each eigenvalue λ, solve (A−λI)x = 0 using Gaussian elimination.
For λ₁ = 3: (A−3I) = [[0,1],[0,−1]] → row reduce → y=0, x free
→ eigenvector x₁ = [1,0]ᵀ
For λ₂ = 2: (A−2I) = [[1,1],[0,0]] → x+y=0 → x=−y, y free
→ eigenvector x₂ = [−1,1]ᵀ
→ eigenvector x₁ = [1,0]ᵀ
For λ₂ = 2: (A−2I) = [[1,1],[0,0]] → x+y=0 → x=−y, y free
→ eigenvector x₂ = [−1,1]ᵀ
3×3 Example
A = [[2,0,0],[1,3,0],[0,0,2]] (upper triangular)
Eigenvalues: diagonal entries → λ=2 (mult. 2), λ=3 (mult. 1)
Eigenvalues: diagonal entries → λ=2 (mult. 2), λ=3 (mult. 1)
Key fact: Eigenvalues of triangular matrix = diagonal entries
Key Properties
- det(A) = product of all eigenvalues: det(A) = λ₁·λ₂·…·λₙ
- trace(A) = sum of all eigenvalues: tr(A) = λ₁+λ₂+…+λₙ
- λ is eigenvalue of A iff λⁿ is eigenvalue of Aⁿ
- A is singular iff 0 is an eigenvalue
Diagonalization
If A has n linearly independent eigenvectors: A = PDP⁻¹
where D = diag(λ₁,…,λₙ) and P = [x₁|x₂|…|xₙ] (eigenvectors as columns)
Then Aⁿ = PDⁿP⁻¹ (efficient computation of matrix powers)
where D = diag(λ₁,…,λₙ) and P = [x₁|x₂|…|xₙ] (eigenvectors as columns)
Then Aⁿ = PDⁿP⁻¹ (efficient computation of matrix powers)