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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)

Lecture 20 - Four Subspaces associated with a Given Matrix

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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