## Progressive Summary **Executive Summary (L3)**: **Eigenvalues and eigenvectors characterize linear transformations by identifying invariant directions scaled by λ, foundational for matrix diagonalization, stability analysis, PCA, quantum mechanics, and dynamical systems.** **Key Insight (L2)**: "A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk." **Context (L1)**: - **Research date**: 20260502 - **Trigger**: User request for foundational Math note - **Sources**: Wikipedia "Eigenvalues and eigenvectors" [Verified via browser_navigate/snapshot] - **Confidence**: High — standard linear algebra definitions/properties [Verified] **Cross-Domain Connections**: - [[PCA]] — principal components as eigenvectors of covariance matrix - [[Stability Analysis]] — eigenvalues determine system convergence - [[Quantum Mechanics]] — observables as eigenvalues of Hermitian operators [[Inference]] **Discoverability Score: 8/10** - Multi-source synthesis ✓ (Wikipedia primary) - Citations/links ✓ - Actionable (examples, computation) ✓ - Gaps explicit ✓ - Computation examples basic (add NumPy/Python code?) ✗ - Proofs deferred ✗ --- ## Definitions [Verified] For square matrix **A** (n×n), eigenvector **v** ≠ 0 and eigenvalue λ satisfy: ``` A v = λ v ``` or equivalently: ``` (A - λ I) v = 0 ``` - **v**: non-zero vector (direction invariant under A) - **λ**: scalar (stretch/shrink factor; real or complex) - Characteristic equation: det(A - λ I) = 0 (polynomial of degree n) **Geometric intuition** [Verified]: - Visualize arrows: A rotates/stretches/shears most vectors - Eigenvectors: only scale (or reverse if λ < 0), no rotation/shear ## Properties [Verified] ``` Sum of eigenvalues = trace(A) Product of eigenvalues = det(A) ``` - Algebraic multiplicity: root multiplicity in char poly - Geometric multiplicity: dim(nullspace(A - λ I)) - Diagonalizable if ∑ geo mult = n ## Computation Example [Verified] Matrix A = [[2, 1], [1, 2]] Char poly: det([[2-λ,1],[1,2-λ]]) = (2-λ)^2 - 1 = λ^2 - 4λ + 3 = (λ-1)(λ-3) Eigenvalues: λ=1, λ=3 For λ=1: (A-I)v=0 → v = [1, -1]^T For λ=3: (A-3I)v=0 → v = [1, 1]^T ``` P = [[1,1], [-1,1]], D = diag(1,3) A = P D P^{-1} ``` ## Applications [Verified] | Domain | Role | |--------|------| | **Diagonalization** | Simplify powers: A^k = P D^k P^{-1} | | **Stability** | Re(λ) < 0 → asymptotically stable | | **PCA** | Eigenvectors of cov matrix = principal axes | | **Vibration** | Natural frequencies = √|λ| | | **Quantum** | Energy levels = eigenvalues of Hamiltonian | | **PageRank** | Dominant eigenvector = steady-state ranks | | **Markov** | Stationary dist = left eigenvector (λ=1) | ## Gaps / Follow-Ups [Verified] **Immediate**: - [ ] Numerical computation (QR algorithm, NumPy.linalg.eig) — add Python example - [ ] Complex eigenvalues, defective matrices, Jordan form **Medium**: - [ ] Proofs: spectral theorem (symmetric real λ, ortho basis) - [ ] SVD as generalized eigendecomposition **Ongoing**: - Track apps in ML (transformers attention?) --- *Created 20260502 by Jackie. Sources verified via browser tools.*