Eigenvectors & Eigenvalues: How a Matrix Acts on Space

A 2×2 matrix A maps every point in the plane somewhere else. Most vectors get rotated and scaled. Eigenvectors are the special directions that only get scaled — they don't rotate. Drag the matrix entries below and watch which directions stay fixed.

unit circle → image ellipse eigenvector directions a sample vector v → Av

Matrix A — adjust the entries below

A =

a₁₁ 2.00

a₁₂ 1.00

a₂₁ 1.00

a₂₂ 1.50

Sample vector v angle θ

How to read this: the orange circle is the set of all unit vectors. A stretches it into the sage ellipse. The sage arrows are the eigenvectors of A — the only directions where the input arrow and its image Av point along the same line. Their lengths are scaled by the corresponding eigenvalues, λ₁ and λ₂. When the eigenvalues are complex (e.g. try the "rotation" preset), there are no real fixed directions — every vector gets rotated, and the eigenvector arrows disappear.