★★☆ Medium

🧮 Eigenvectors & Eigenvalues

Adjust any 2×2 matrix and see its eigenvectors — the special directions that a linear transformation only stretches, never rotates. Watch the characteristic polynomial solve for λ, the unit circle deform into an ellipse, and eigenvectors align with its principal axes.

î image ĵ image Unit circle image λ₁ eigenvec λ₂ eigenvec

📐 Matrix A

A[0,0] = a

2.0

A[0,1] = b

1.0

A[1,0] = c

0.0

A[1,1] = d

1.0

🎯 Presets

λ Eigenvalues

λ₁ —

λ₂ —

v₁ = —

v₂ = —

📊 Properties

trace:—

det:—

Δ =—

Characteristic polynomial:
det(A − λI) = 0
λ² − tr(A) λ + det(A) = 0

Av = λv (v ≠ 0)
Eigenvectors span the eigenspace.
For symmetric A: eigenvectors are orthogonal.

Key ideas

Eigenvectors are the vectors v that satisfy A v = λ v for a scalar λ (the eigenvalue). Geometrically, they are the directions the transformation only stretches or flips — never rotates. Every point on the green/purple dashed lines is an eigenvector.

The characteristic polynomial λ² − tr(A)λ + det(A) = 0 gives the eigenvalues. Its discriminant Δ = tr² − 4 det determines the type: Δ > 0 → two distinct real eigenvalues; Δ = 0 → repeated; Δ < 0 → complex conjugates (pure rotation has no real eigenvectors).

The yellow ellipse is the image of the unit circle under A. Its semi-axes are exactly the eigenvectors scaled by |λ|. Related: Matrix Transformations →