Eigenvalues and eigenvectors

An eigenvector of a matrix is a vector that only gets stretched (not turned) by the matrix; the stretch factor is its eigenvalue. They reveal the natural directions of a transformation and appear everywhere from Google search to quantum physics.

Key facts

  • Av = λv for eigenvector v and eigenvalue λ.
  • Find λ from the characteristic equation det(A − λI) = 0.
  • For 2 × 2: λ² − (trace)λ + det = 0.
  • Then solve (A − λI)v = 0 for each λ.

Worked examples

Example 1

Find the eigenvalues of [[4, 1], [2, 3]].

  1. Trace = 7, det = 12 − 2 = 10.
  2. λ² − 7λ + 10 = 0 (λ − 2)(λ − 5) = 0
  3. λ = 2 or 5

Answer: 2 and 5