Linear Algebra

Eigenvectors & Eigenvalues

Most vectors get knocked off their line when a matrix acts on them. Eigenvectors are the special directions that only stretch. Spin the input around and hunt for the moment the output lines up.

Matrix A

a 2.00

b 1.00

c 1.00

d 2.00

Presets

Transform amount applied

identityfull A

Input angle 0°

Show the warped grid

Eigenvalues—

|Av| / |v|—

v ∠ Av—

What to observe

Drag the blue input vector or sweep the angle. The orange outputAv swings around too, but at a different rate: it is not just a rotation of the input.
When v ∠ Av hits 0° (or 180°) the two line up. ThereAv = λv: you have found an eigenvector, and the stretch factor is the eigenvalue λ.
Try the presets. Symmetric has perpendicular eigenvectors,Shear has only one, and Rotation has none in the real plane, its eigenvalues are complex.

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