Eigenvectors as Fixed Directions
Fundamentals of AI/ML · Prof. Parikshit Pareek & Prof. Jitin Singla · IIT Roorkee
A vector v is an eigenvector of A if multiplying by A doesn't change its direction — only its length. The scaling factor λ is the eigenvalue. Drag the blue arrow v around: usually Av points somewhere else. But when v lies on one of the dashed eigen-lines, Av stays on the same line. Press Iterate to watch power iteration drag any starting v toward the dominant eigenvector.
A v = λ v
Matrix A (drag sliders)
Presets:
vector v (drag tip)
A v
eigen-directions
iteration trail
Vector data
v = (—) ‖v‖ = —
Av = (—) ‖Av‖ = —
‖Av‖ / ‖v‖ = —
Angle between v and Av: —
Eigenvector test: not aligned
Matrix A
tr(A) = —, det(A) = —
Eigendecomposition
—
What to notice during class
- The eigen-lines are special. Drag v to land on a green dashed line; Av snaps to that same line. Off the line, Av rotates away.
- Symmetric A has perpendicular eigenvectors. Try the Symmetric preset: the two green lines are at right angles. Spectral theorem made visible.
- Shear (A = [[1,1],[0,1]]) has only one eigen-line. The y-axis. Every other vector picks up a horizontal component. Defective matrix — cannot be diagonalized.
- Rotation has complex eigenvalues. Try the Rotation preset: no real direction is preserved (except trivially when the rotation is by 0 or π). The eigenvectors live in ℂ².
- Power iteration always works (almost). Click "Power iteration" with any starting v. After a few steps it lands on the dominant eigen-line. This is how Google's PageRank was computed.
- tr(A) = λ1 + λ2, det(A) = λ1λ2. Quick sanity check — useful for ETE problems.