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Numerical Linear Algebra


  • Basic Matrix Operations : - Matrix–Matrix multiplication. Operation counts; 
  • Linear Spaces and Mappings:  Range and Null spaces. Rank. Scalar Product, Vector and Matrix Norms
  • Direct methods for solving linear algebraic systems: The Inverse, Rank, The LU Decomposition, Cholesky Decomposition, Sensitivity, Condition number, Residual; Least Squares Problems:  Normal Equations, Data fitting, Gram-Schmitt orthogonalization, The QR decomposition, Projections.
  • The QR Decomposition: Householder Reflections, Avoiding the Q; Structured Least Squares problems: Givens Rotations and Row updating; Applications: Data fitting, Tikhonov Regularization
  • The Eigenvalue Decomposition: Definitions and Basics, Localization of Eigenvalues. Sensitivity; Computing Eigenvalues: Rayleigh Quotient, The Power iteration, Inverse Iteration, Decoupling, Similarity Transformations, The Schur Decomposition, Hessenberg Decomposition, The QR Algorithm, Shifts; Applications: Google PageRank.
  • The Singular value decomposition: Definition, Fundamental subspaces, Linear systems of equations and least squares problems, Low rank approximation, Numerical Rank, Classification of hand-written digits; Applications of the SVD: remote sensing examples, Image deblurring.
  • Sparse Matrices and Sparse Linear Systems: Compress-Sparse-Row (CSR) storage scheme, Example: Sparse Matrix from discretization of a PDE, Classic Iterative Methods and Convergence results; Projection Methods for Linear Systems: Definition of a general projection method, Optimality results, Minimal Residual method; Krylov subspace methods: Krylov subspaces, Generalized Minimal Residual (GMRES), Conjugate Gradient Method (CG); Sparse Least Squares problems: Application in data mining