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Course11110215258 lectures

Matrix Computation and its applications

Prof. Mani Mehra and Prof. Vivek Kumar Aggarwal · IIT Delhi

Start with lecture 1 At NPTEL

About

This course deals with applications of matrices to a wide range of areas of engineering and science. Initial some basics of linear algebra is discussed followed by matrix norms and sensitivity and condition number of the matrices. In this course , we will also discuss psudo-inverse of a matrix of any dimension (m x n) using Moore- Penrose theorem. Singular value decomposition of a general matrix is also discussed along with applications. Householder transformations, QR factorization will also be covered.

Intended audience: UG/PG

Prerequisites: Some knowledge of matrix theory

Lectures

Week 1

  1. Binary Operation and Groups YouTube
  2. Vector Spaces YouTube
  3. Some Example of Vector Spaces YouTube
  4. Some Example of Vector Spaces (contd.) YouTube
  5. Subspace of a Vector Space YouTube
  6. Spanning Set YouTube

Week 2

  1. Properties of Subspace YouTube
  2. Properties of Subspace (contd.) YouTube
  3. Linearly independent and dependent vectors YouTube
  4. Linearly independent and dependent vectors (contd.) YouTube
  5. Properties of linearly independent and dependent vectors YouTube
  6. Properties of linearly independent and dependent vectors (contd.) YouTube

Week 3

  1. Basis and Dimension of a Vector Space YouTube
  2. Examples of Basis and standard basis of a vector space YouTube
  3. Linear Functions YouTube
  4. Range space of a matrix and row reduced echelon form YouTube
  5. Row equivalent matrices YouTube
  6. Row equivalent matrices (contd.) YouTube

Week 4

  1. Null Space of a Matrix YouTube
  2. Four subspaces associated with a given matrix YouTube
  3. Four subspaces associated with a given matrix (contd.) YouTube
  4. Linear Independence of the rows and columns of a matrix YouTube
  5. Application of diagonal dominant matrices YouTube
  6. Application of Zero Null space: Interpolating polynomial and wronskian matrix YouTube

Week 5

  1. Characterization of basis of a vector space and its subspaces YouTube
  2. Coordinate of a vector with respect to ordered basis YouTube
  3. Examples of different subspaces of a vector space of polynomials having degree less than or equal to 3 YouTube
  4. Linear transformation YouTube
  5. Properties of linear transformation YouTube
  6. Determining linear transformation on a vector space by its value on the basis element YouTube

Week 6

  1. Range space and null space of a linear transformation YouTube
  2. Rank and nuility of a linear transformation YouTube
  3. Rank-nuility theorem YouTube
  4. Application of rank-nuility theorem and inverse of a linear transformation YouTube
  5. Matrix associated with linear transformation YouTube
  6. Matrix representation of a linear transformation relative to ordered bases YouTube

Week 7

  1. Matrix representation of a linear transformation relative to ordered bases (contd.) YouTube
  2. Linear map associated with a matrix YouTube
  3. Similar matrices and diagonalisation of matrix YouTube
  4. Orthonormal bases of a vector space YouTube

Week 8

  1. Gram-Schmidt orthogonalisation process YouTube
  2. QR factorisation YouTube
  3. Inner Product Spaces YouTube
  4. Inner product on different real vector spaces and basis of complex vector space YouTube

Week 9

  1. Inner product on complex vector spaces and Cauchy-Schwarz inequality YouTube
  2. Norm of a vector YouTube
  3. Matrix norm YouTube
  4. Sensitivity analysis of a system of linear equations YouTube

Week 10

  1. Orthogonality of the four subspaces associated with a Matrix YouTube
  2. Best approximation : Least square method YouTube
  3. Best approximation : Least square method (contd.) YouTube
  4. Jordan-Canonical Form YouTube

Week 11

  1. Some examples on the Jordan form of a given matrix and generalised eigenvectors YouTube
  2. Singular value decomposition (SVD) theorem YouTube
  3. Matlab / Octave code for solving SVD YouTube

Week 12

  1. Pseudo-inverse/Moore-Penrose inverse YouTube
  2. Householder transformation YouTube
  3. Matlab/Octave code for Householder transformation YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Delhi
Department
Mathematics
Taught by
Prof. Mani Mehra, Prof. Vivek Kumar Aggarwal
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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