Supplementary notes

• LU and PLU factorizations

  Derivation of the LU and PLU factorizations from Gaussian elimination

• Schur decomposition

  Derivation of the Schur decomposition

• Singular value decomposition

  Derivation and properties of the singular value decomposition

• Projections and least squares

  Orthogonal projections, least squares problems, and the pseudoinverse

• QR factorization

  Derivations of the QR factorization using Gram–Schmidt orthogonalization, Householder reflections, and Givens rotations

• Linear stationary iterative methods

  Linear stationary iterative (splitting) methods and convergence theorems

• Projection methods for linear systems

  Projection methods for linear systems, error and residual projection methods

• Krylov subspace methods

  Krylov subspace methods for linear systems (e.g., GMRES, CG)

• Conjugate gradient method

  Derivation of the conjugate gradient method as a Krylov subspace method