📐 Eigenvalues & Eigenvectors — Linear Transformation Visualiser
Visualise how a 2×2 matrix A transforms vectors in the plane. Eigenvectors (Av = λv) are special directions that only scale — they stay on their own span. Watch the unit circle transform into an ellipse, the grid warp, and see eigenvectors highlighted in red and blue.
Mathematics
For matrix A, eigenvectors satisfy Av = λv. Eigenvalues are found from det(A − λI) = 0 → λ² − tr(A)·λ + det(A) = 0, giving λ = (tr ± √(tr²−4det))/2. When the discriminant is negative, eigenvalues are complex (rotation-like behaviour). Real symmetric matrices always have real orthogonal eigenvectors. The unit circle under A maps to an ellipse whose semi-axes align with the eigenvectors and have length |λ₁| and |λ₂|. The determinant equals λ₁·λ₂ and gives the signed area scaling.