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September 23, 2023

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Description

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)

Course Curriculum

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

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