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Numerical Linear Algebra

Numerical Linear Algebra. Instructors: Dr. P. N. Agrawal and Dr. D. N. Pandey, Department of Mathematics, IIT Roorkee. This course covers lessons in basics of matrix algebra, computer arithmetic, conditioning and condition number, stability of numerical algorithms, vector and matrix norms, convergent matrices, stability of nonlinear 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. (from nptel.ac.in)

Lecture 45 - Algebraic and Geometric Properties of Matrices using SVD


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Lecture 01 - Matrix Operations and Types of Matrices
Lecture 02 - Determinant of a Matrix
Lecture 03 - Rank of a Matrix
Lecture 04 - Vector Space
Lecture 05 - Vector Space (cont.)
Lecture 06 - Linear Dependence and Independence
Lecture 07 - Bases and Dimensions
Lecture 08 - Bases and Dimensions (cont.)
Lecture 09 - Linear Transformation
Lecture 10 - Linear Transformation (cont.)
Lecture 11 - Orthogonal Subspaces
Lecture 12 - Row Space, Column Space and Null Space
Lecture 13 - Eigenvalues and Eigenvectors
Lecture 14 - Eigenvalues and Eigenvectors (cont.)
Lecture 15 - Diagonalizable Matrices
Lecture 16 - Orthogonal Sets
Lecture 17 - Gram Schmidt Orthogonalization and Orthonormal Bases
Lecture 18 - Introduction to MATLAB
Lecture 19 - Sign Integer Representation
Lecture 20 - Computer Representation of Numbers
Lecture 21 - Floating Point Representation
Lecture 22 - Round-off Error
Lecture 23 - Error Propagation in Computer Arithmetic
Lecture 24 - Addition and Multiplication of Floating Point Numbers
Lecture 25 - Conditioning and Condition Numbers
Lecture 26 - Conditioning and Condition Numbers (cont.)
Lecture 27 - Stability of Numerical Algorithms
Lecture 28 - Stability of Numerical Algorithms (cont.)
Lecture 29 - Vector Norms
Lecture 30 - Vector Norms (cont.)
Lecture 31 - Matrix Norms
Lecture 32 - Matrix Norms (cont.)
Lecture 33 - Convergent Matrices
Lecture 34 - Convergent Matrices (cont.)
Lecture 35 - Stability of Nonlinear System
Lecture 36 - Condition Number of A Matrix: Elementary Properties
Lecture 37 - Sensitivity Analysis
Lecture 38 - Sensitivity Analysis (cont.)
Lecture 39 - Residual Theorem
Lecture 40 - Nearness to Singularity
Lecture 41 - Estimation of Condition Number
Lecture 42 - Singular Value Decomposition of a Matrix
Lecture 43 - Singular Value Decomposition of a Matrix (cont.)
Lecture 44 - Orthogonal Projections
Lecture 45 - Algebraic and Geometric Properties of Matrices using SVD
Lecture 46 - SVD and their Applications
Lecture 47 - Perturbation Theorem for Singular Values
Lecture 48 - Outer Product Expansion of a Matrix
Lecture 49 - Least Square Solutions
Lecture 50 - Least Square Solutions (cont.)
Lecture 51 - Householder Matrices
Lecture 52 - Householder Matrices and their Applications
Lecture 53 - Householder QR Factorization
Lecture 54 - Householder QR Factorization (cont.)
Lecture 55 - Basic Theorems on Eigenvalues and QR Method
Lecture 56 - Power Method
Lecture 57 - Rate of Convergence of Power Method
Lecture 58 - Applications of Power Method with Shift
Lecture 59 - Jacobi Method
Lecture 60 - Jacobi Method (cont.)