Home▸Mathematics▸Eigenvalues and Eigenvectors: The Directions a Matrix Refuses to Rotate

📐 Eigenvalues and Eigenvectors: The Directions a Matrix Refuses to Rotate

Interactive 3D vector field where adjusting a transformation matrix's entries shows most vectors rotating while eigenvectors stretch or shrink along their own direction, highlighted distinctly.

Mathematics3DModerate60 FPS

A 3D field of vectors on a unit circle is transformed live by a 2×2 matrix: most vectors visibly rotate off their original line, while the matrix's eigenvectors — highlighted in gold and pink — only stretch or shrink along their own direction.

🔬 What It Demonstrates

Every sample vector is colored by how far the transformation rotates it: blue means almost no rotation, orange means a large swing. The two directions that stay blue no matter what are the real eigenvectors, scaled by their eigenvalues.

🎮 How to Use

Drag the a, b, c, d sliders to build any 2×2 matrix, or jump to a preset (saddle, stretch, shear, rotation, reflection). Scrub "Transform amount" to morph from the identity to the full transformation, or let it animate automatically.

💡 Did You Know?

When a matrix's discriminant is negative its eigenvalues become complex numbers — geometrically that means every real direction gets rotated, which is exactly what a pure rotation matrix does.

⚙ Under the hood

Interactive 3D vector field where adjusting a transformation matrix's entries shows most vectors rotating while eigenvectors stretch or shrink along their own direction, highlighted distinctly.

eigenvalueseigenvectorslinear-algebralinear-transformationsmatricesmathematics

3D · Three.js / WebGL renderer · 60 FPS target · runs fully client-side, no install