Eigenvectors

Directions a matrix only stretches

An interactive lesson with 7 steps, about 6 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• An eigenvector keeps its line: $A\mathbf v = \lambda\mathbf v$.
• Eigenvalues solve $\det(A - \lambda I) = 0$. They sum to the trace and multiply to the determinant.
• $A^n\mathbf v = \lambda^n\mathbf v$: powers of a matrix become powers of numbers.

Steps

1. Hunting for special directions
2. Eigenvectors and eigenvalues
3. Compute eigenvalues
4. No special directions?
5. Trace and determinant
6. Repeated application
7. Everything swings to the eigenvector