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Eigenvalues & Eigenvectors

See how a 2×2 matrix transforms space — eigenvectors stay collinear, eigenvalues reveal the stretching factor

Linear Algebra Mathematics Matrix Theory PCA

a: 2.0 b: 1.0 c: 0.0 d: 1.0

Show transformed grid Unit circle → ellipse Eigenvectors Animate transition

λ₁ = — λ₂ = — trace = — det = — Type: —

📐 Eigenvalues & Eigenvectors

A vector v is an eigenvector of matrix A if the transformation only scales it — it does not rotate:
A·v = λ·v — where scalar λ is the corresponding eigenvalue.

To find eigenvalues, solve the characteristic equation:
det(A − λI) = 0 ⟹ λ² − tr(A)·λ + det(A) = 0
giving λ = (tr ± √(tr² − 4·det)) / 2

Geometrically: the unit circle maps to an ellipse whose semi-axes point along the eigenvectors with lengths equal to |λ|. Real distinct eigenvalues → stretching/compression along two axes. Complex eigenvalues → rotation + spiral. Repeated eigenvalues → uniform scaling or shear.

Eigenvalues are central to PCA, quantum mechanics (energy levels), graph theory (spectral graph theory), and stability analysis of ODEs.