For A: V → V, a scalar λ and nonzero v with Av = λv. The characteristic polynomial det(A − λI) = 0 gives spectrum. Basis of spectral theory, PCA, and quantum observables.
Eigenvalue / eigenvector
Related concepts
- Linear map
- Spectral theorem
- Jordan canonical form
- Spectral graph theory
- Cayley–Hamilton theorem
- Schur decomposition
- Sturm–Liouville theory
- Morse theory (critical points)
- Spectral theorem for bounded self-adjoint operators
- Small oscillations (normal modes)
- Ab initio electronic-structure methods
- Configuration interaction (CI)
- Hückel (4n+2) rule: planar conjugated monocycle aromatic iff π-count mod 4 = 2
- Walsh diagrams (AH₂, AH₃)
- Perovskite solar cells
- Tight-binding model
- Spin-state crossover in metalloproteins
- PCA (Pearson 1901 / Hotelling 1933)
- R₀ via next-generation matrix: R₀ = ρ(K) = spectral radius; 2×2 NGM = √(a12·a21)
- May food-web stability (Jacobian random-matrix)