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Eigenvalues are scalar values that characterize how a matrix transforms space. If Av = λv (where v ≠ 0), then λ is an eigenvalue and v is an eigenvector. Finding eigenvalues means solving the characteristic equation det(A − λI) = 0.
Characteristic Equation (λ)
det(A − λI) = 0
Subtract λ from each diagonal entry of matrix A. For A = [[4, 1], [2, 3]]: A − λI = [[4−λ, 1], [2, 3−λ]].
A − λIFor a 2×2 matrix [[a,b],[c,d]], det = ad − bc. det(A − λI) = (4−λ)(3−λ) − (1)(2) = λ² − 7λ + 10.
det(A − λI) = (4−λ)(3−λ) − 2Set det = 0: λ² − 7λ + 10 = 0. Factor: (λ−5)(λ−2) = 0. Eigenvalues: λ₁ = 5, λ₂ = 2.
λ² − 7λ + 10 = 0Check: trace(A) = 4 + 3 = 7 = λ₁ + λ₂ = 5 + 2 ✓. det(A) = 12 − 2 = 10 = λ₁ · λ₂ = 5 × 2 ✓.
tr(A) = Σλᵢ, det(A) = ΠλᵢEigenvalues are used in Google PageRank (the dominant eigenvalue determines page importance), quantum mechanics (energy levels are eigenvalues of the Hamiltonian), structural engineering (natural vibration frequencies), data science (PCA uses eigenvalues for dimensionality reduction), and stability analysis in differential equations.
Don't forget to subtract λ from EVERY diagonal entry. For 3×3 matrices, the characteristic polynomial is cubic — use the rational root theorem or numerical methods. Complex eigenvalues indicate rotation in the transformation.
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