Numerical Linear Algebra

This course is a basic course offered to UG/PG students of Engineering/Science background. It contains basics of matrix algebra, computer arithmetic, conditioning and condition number, stability of numerical algorithms, vector and matrix norms, convergent matrices, stability of non-linear systems, sensitivity analysis, singular value decomposition (SVD), algebraic and geometric properties of SVD, least square solutions, Householder matrices and applications, QR method, Power method and applications, Jacobi method for finding the eigenvalues of a given matrix. This course has tremendous applications in diverse fields of Engineering and Sciences such as control theory, image processing, numerical analysis and dynamical systems etc.INTENDED AUDIENCE : UG and PG students of technical institutions/ universities/collegesPRE-REQUISITES : NoneINDUSTRY SUPPORT : None

Week 1: Matrix operations and type of matrices, Determinant of a Matrix, Rank of a matrix, Vector Space-I, Vector Space-IIWeek 2: Linear dependence and independence, Bases and Dimensions – I, Bases and Dimension - II, Linear Transformation - I, Linear Transformation - IIWeek 3: Orthogonal subspaces, Row space, column space and null Space, Eigenvalues and Eigenvectors-I, Eigenvalues and Eigenvectors-II, Diagonalizable MatricesWeek 4: Orthogonal Sets, Gram Schmidt orthogonalization and orthonormal bases, Introduction to Matlab, Sign integer representation Computer representation of numbersWeek 5: Floating point representation, Round-off error, Error propagation in computer arithmetic, Addition and multiplication of floating point numbers, Conditioning and condition numbers-IWeek 6: Conditioning and condition numbers-II, Stability of numerical algorithms-I, Stability of numerical algorithms-II, Vector norms - I, Vector norms - IIWeek 7: Matrix Norms - I, Matrix Norms-II, Convergent Matrices - I, Convergent Matrices - II, Stability of non-linear systemWeek 8: Condition number of a matrix: Elementary properties, Sensitivity analysis-I, Sensitivity analysis-II, Residual theorem, Nearness to singularityWeek 9: Estimation of the condition number, Singular value decomposition of a matrix – I, Singular value decomposition of a matrix - II, Orthogonal Projections, Algebraic and geometric properties of matrices using SVDWeek 10: SVD and their applications, Perturbation theorem for singular values, Outer product expansion of a matrix, Least square solutions-I, Least square solutions-IIWeek 11: Psudeo - inverse and least square solution, Householder matrices and their applications, Householder QR factorization –I, Householder QR factorization –II, Basic theorems on eigenvalues and QR methodWeek 12: Power method, Rate of convergence of Power method, Applications of Power method with shift, Jacobi method-I, Jacobi method-II