Finding Eigenvectors
You've found the eigenvalues. Now for each one, solve (A − λI)v = 0 to find the special directions — the eigenvectors that span the eigenspace.
The Null Space of (A − λI)
The eigenspace E_λ is all solutions to (A − λI)v = 0. It's a subspace — always including zero, but eigenvectors are the nonzero elements.
Four Steps to Eigenvectors
- Form (A − λI): subtract λ from the diagonal
- Row-reduce to RREF — always at least one free variable
- Express pivot variables in terms of free variables
- Each free variable → one basis eigenvector
Eigenvectors for λ = 3
Eigenvectors for λ = 2
How Many Directions?
The geometric multiplicity of λ is dim(E_λ) — the number of independent eigenvectors. It's always between 1 and the algebraic multiplicity.
When Geo < Alg. Multiplicity
If geometric < algebraic multiplicity, the eigenvalue is defective. Example: [[2, 1], [0, 2]] has λ = 2 with alg. mult. 2 but only one eigenvector [1, 0]ᵀ.
Distinct Eigenvalues → Independence
Eigenvectors from distinct eigenvalues are always linearly independent. An n×n matrix with n distinct eigenvalues has n independent eigenvectors — perfect for diagonalization.
- Proof: apply A, subtract, use distinctness
- Generalizes to any number of eigenvalues
- Foundation for diagonalization theorem
Rotation in Disguise
- Real matrices can have complex eigenvalues
- Always appear in conjugate pairs: λ and λ̄
- Correspond to rotation-and-scaling behavior
- Plane (2×2) rotation eigenvalues: e^{±iθ}
- No real eigenvectors for a plane rotation, θ ≠ 0 or π
- In 3D a rotation does fix its axis — a real eigenvector at λ = 1
M4-L3: Diagonalization
You can find eigenvalues and eigenvectors. Next: use them to diagonalize A = PDP⁻¹, unlocking fast matrix powers and deep structural insight.