# InfoCoBuild

## Matrix Computation and its Applications

Matrix Computation and its Applications. Instructor: Prof. Vivek Kumar Aggarwal, Department of Mathematics, IIT Delhi. This course deals with applications of matrices to a wide range of areas of engineering and science. Some basics of linear algebra are discussed followed by matrix norms and sensitivity and condition number of the matrices. The course continues to discuss topics: linear systems, Jacobi, Gauss-Seidel and successive over relaxation methods, LU decompositions, Gaussian elimination with partial pivoting, Banded systems, positive definite systems, Cholesky decomposition - sensitivity analysis, Gram-Schmidt orthonormal process, Householder transformation, QR factorization, stability of QR factorization. Solution of linear least squares problems, normal equations, singular value decomposition (SVD), Moore-Penrose inverse, rank deficient least squares problems, sensitivity analysis of least squares problems, sensitivity of eigenvalues and eigenvectors. (from nptel.ac.in)

 Introduction

 Lecture 01 - Binary Operation and Groups Lecture 02 - Vector Spaces Lecture 03 - Some Examples of Vector Spaces Lecture 04 - Some Examples of Vector Spaces (cont.) Lecture 05 - Subspace of a Vector Space Lecture 06 - Spanning Set Lecture 07 - Properties of Subspace Lecture 08 - Properties of Subspace (cont.) Lecture 09 - Linearly Independent and Dependent Vectors Lecture 10 - Linearly Independent and Dependent Vectors (cont.) Lecture 11 - Properties of Linearly Independent and Dependent Vectors Lecture 12 - Properties of Linearly Independent and Dependent Vectors (cont.) Lecture 13 - Basis and Dimension of a Vector Space Lecture 14 - Examples of Basis and Dimension of a Vector Space Lecture 15 - Linear Functions Lecture 16 - Range Space of a Matrix and Row Reduced Echelon Form Lecture 17 - Row Equivalent Matrices Lecture 18 - Row Equivalent Matrices (cont.) Lecture 19 - Null Space of a Matrix Lecture 20 - Four Subspaces associated with a Given Matrix Lecture 21 - Four Subspaces associated with a Given Matrix (cont.) Lecture 22 - Linear Independence of the Rows and Columns of a Matrix Lecture 23 - Application of Diagonal Dominant Matrices Lecture 24 - Application of Zero Null Space: Interpolating Polynomial and Wronskian Matrix Lecture 25 - Characterization of Basis of a Vector Space and its Subspaces Lecture 26 - Coordinate of a Vector with respect to Ordered Basis Lecture 27 - Examples of Different Subspaces of a Vector Space of Polynomials Lecture 28 - Linear Transformation Lecture 29 - Properties of Linear Transformation Lecture 30 - Determining Linear Transformation on a Vector Space by its Value on the Basis Element Lecture 31 - Range Space and Null Space of a Linear Transformation Lecture 32 - Rank and Nullity of a Linear Transformation Lecture 33 - Rank-Nullity Theorem Lecture 34 - Application of Rank-Nullity Theorem and Inverse of a Linear Transformation Lecture 35 - Matrix Associated with Linear Transformation Lecture 36 - Matrix Representation of a Linear Transformation Relative to Ordered Bases Lecture 37 - Matrix Representation of a Linear Transformation Relative to Ordered Bases (cont.) Lecture 38 - Linear Map associated with a Matrix Lecture 39 - Similar Matrices and Diagonalization of Matrix Lecture 40 - Orthonormal Bases of a Vector Space Lecture 41 - Gram-Schmidt Orthogonalization Process Lecture 42 - QR Factorization Lecture 43 - Inner Product Spaces Lecture 44 - Inner Product on Different Real Vector Spaces and Basis of Complex Vector Space Lecture 45 - Inner Product on on Complex Vector Spaces and Cauchy-Schwarz Inequality Lecture 46 - Norm of a Vector Lecture 47 - Matrix Norm Lecture 48 - Sensitivity Analysis of a System of Linear Equations Lecture 49 - Orthogonality of the Four Spaces associated with a Matrix Lecture 50 - Best Approximation: Least Square Method Lecture 51 - Best Approximation: Least Square Method (cont.) Lecture 52 - Jordan-Canonical Form Lecture 53 - Some Examples on the Jordan Form of a Given Matrix and Generalized Eigenvectors Lecture 54 - Singular Value Decomposition Theorem Lecture 55 - MatLab/Octave Code for Solving SVD Lecture 56 - Pseudo-Inverse/Moore-Penrose Inverse Lecture 57 - Householder Transformation Lecture 58 - MatLab/Octave Code for Householder Transformation

 References Matrix Computation and its Applications Instructor: Prof. Vivek Kumar Aggarwal, Department of Mathematics, IIT Delhi. This course deals with applications of matrices to a wide range of areas of engineering and science.