faheem.

Linear Algebra

Inner products, decompositions, spectral theory. This is the background the other notes assume you have.

  1. 01Vector Spaces and Inner ProductsThe axioms, then an inner product. Carrying both over to function spaces is what most of the physics runs on.
  2. 02Eigenvalues and the Spectral TheoremWhen is a matrix guaranteed a full set of eigenvectors? The conditions are less generous than you would hope.
  3. 03Singular Value DecompositionThe one decomposition that works on any matrix whatsoever. What each of its three factors is doing.
  4. 04Quadratic Forms and DefinitenessRead a symmetric matrix as a function, and the spectral theorem sorts it by sign.
  5. 05The Matrix ExponentialSolving a coupled linear system by exponentiating its matrix, computed through the eigenbasis.