📐 Eigenvalues and Eigenvectors

About this simulation

A matrix M takes every vector in the plane and sends it somewhere new — usually rotating it as well as stretching it. But a handful of special directions, the eigenvectors, are mapped onto themselves: the matrix only scales them, by a factor called the eigenvalue (λ), and never rotates them off their own line. This simulation draws a whole field of vectors on a unit circle and shows, in real time, which ones the matrix "refuses to rotate."

How it works

• Each thin line is a sample vector on the unit circle, colored by how much its direction changes under M: blue means almost no rotation, orange means a large swing.
• The two thick highlighted rays are the exact real eigenvectors (when they exist), extended through the origin and scaled by their eigenvalue.
• The dashed ring is the original unit circle; it deforms into an ellipse under the transformation — its long and short axes align with the eigenvectors exactly when M is symmetric.
• When the discriminant of the characteristic equation is negative, the eigenvalues are complex: there is no real direction left unrotated, and M behaves like a pure rotation (plus scaling).

Controls

• a, b, c, d — the four entries of the 2×2 matrix [[a,b],[c,d]].
• Transform amount — scrub from the untransformed field (0%) to the fully transformed field (100%).
• Preset — jump to classic examples: a saddle, a symmetric stretch, a shear, a rotation, a reflection.
• Animate transform — continuously sweeps the transform amount back and forth.
• Show unit circle / grid — toggles the reference circle, ellipse and floor grid.

Did you know?

Eigenvectors and eigenvalues aren't just a linear-algebra exercise — they describe the natural vibration modes of bridges, the principal axes of stress in materials, the stable population structures in ecology, and the ranking vector behind Google's original PageRank algorithm.