Labs / Mathematics
Eigenvectors
A 2×2 matrix picks up the blue vector v and moves it to a new place, M·v. For most directions the arrow twists to point somewhere new — but along a few special directions the matrix only stretches or flips the vector without turning it at all. Sweep varound and hunt for those directions: they are the matrix's eigenvectors, and the stretch factors are its eigenvalues.
eigenvalues 1.55, 0.65v ∠ 20°stretch 1.00×turn 0°
What to try
- Drag v slowly all the way around. How many directions make the “turn” readout snap to 0° or 180°?
- When the two arrows line up, does the stretch factor match one of the eigenvalues in the readout?
- Turn on “Iterate Mⁿv” from a random start. Which eigenvector does the vector drift toward, and why that one?
- Load the Rotation preset. Why can no real direction ever stay parallel — where did the eigenvectors go?
- Set b and c equal (a symmetric matrix). What's special about the angle between the two eigenvector lines?