Finding Eigenvectors

Linear Algebra for Machine Learning · Axiom Academy

Computing eigenvectors from eigenvalues Once you have the eigenvalues, finding the eigenvectors is a matter of solving a system of linear equations. For each eigenvalue λ, you solve: Example: Using Our Previous Matrix From the previous example, we found eigenvalues λ₁ = 6 and λ₂ = 4 for the matrix: Finding the Eigenvector for λ₁ = 6 This gives us two equations: -x - y = 0 and -x - y = 0 (they're the same!) So y = -x. We can choose x = 1, giving y = -1: Finding the Eigenvector for λ₂ = 4 This gives us: x - y = 0, so x = y. Choosing x = 1, y = 1: The Null Space: Finding eigenvectors means finding the null space of (A - λI) Free Variables: If there are free variables, you can have multiple linearly independent eigenvectors Scaling: Eigenvectors are only defined up to scalar multiplication Dimension: The eigenspace (null space) has dimension at least 1

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.