Math basics › Linear algebra
Eigenvalues and eigenvectors
What they are
An eigenvector v only gets stretched by A, not turned: Av = λv. The stretch factor λ is its eigenvalue.
Finding the eigenvalues
Solve det(A − λI) = 0. For a 2 × 2 matrix this is a quadratic in λ.
Then the eigenvectors
For each λ, solve (A − λI)v = 0. There should be a free variable; if the only answer is v = 0, check λ again.
Quick checks
The eigenvalues add up to the trace (the sum down the diagonal) and multiply to the determinant.