Matrix Theory C4YB1RYDB2

URI	`m2://res/C4YB1RYDB2`
Kind	course
Creators	Chandra Murthy (instructor)
Date	2021-05-22
Scope	in
Ids	course 04dese585:111108157

Expressions

e1: edition

Language	en
Date	2021-05-22

m1: page, IISc Bangalore (record). nptel 111108157
m2: page, 2021-05-22 (record). nptel 111108157

Contents

part 1: Matrix Theory
session 1: Course introduction and properties of matrices
session 2: Vector spaces
session 3: Basis, dimension
session 4: Linear transforms
session 5: Fundamental subspaces of a matrix
session 6: Fundamental theorem of linear algebra
session 7: Properties of rank
session 8: Inner product
session 9: Gram-schmidt algorithm
session 10: Orthonormal matrices definition
session 11: Determinant
session 12: Properties of determinants
session 13: Introduction to norms and inner products
session 14: Vector norms and their properties,
session 15: Applications and equivalence of vector norms
session 16: Summary of equivalence of norms
session 17: Dual norms
session 18: Properties and examples of dual norms
session 19: Matrix norms
session 20: Matrix norms: Properties
session 21: Induced norms
session 22: Induced norms and examples
session 23: Spectral radius
session 24: Properties of spectral radius
session 25: Convergent matrices, Banach lemma
session 26: Recap of matrix norms and Levy-Desplanques theorem
session 27: Equivalence of matrix norms and error in inverses of linear systems
session 28: Errors in inverses of matrices
session 29: Errors in solving systems of linear equations
session 30: Introduction to eigenvalues and eigenvectors
session 31: The characteristic polynomial
session 32: Solving characteristic polynomials, eigenvectors properties
session 33: Similarity
session 34: Diagonalization
session 35: Relationship between eigenvalues of BA and AB
session 36: Eigenvector and principle of biorthogonality
session 37: Unitary matrices
session 38: Properties of unitary matrices
session 39: Unitary equivalence
session 40: Schur's triangularization theorem
session 41: Cayley-Hamilton theorem
session 42: Uses of cayley-hamilton theorem and diagonalizability revisited
session 43: Normal matrices: Definition and fundamental properties
session 44: Fundamental properties of normal matrices
session 45: QR decomposition and canonical forms
session 46: Jordan canonical form
session 47: Determining the Jordan form of a matrix
session 48: Properties of the Jordan canonical form (part 1)
session 49: Properties of the Jordan canonical form (part 2)
session 50: Properties of convergent matrices
session 51: Polynomials and matrices
session 52: Other canonical forms and factorization of matrices: Gaussian elimination & LU factorization
session 53: LU decomposition
session 54: LU decomposition with pivoting
session 55: Solving pivoted system and LDM decomposition
session 56: Cholesky decomposition and uses
session 57: Hermitian and symmetric matrix
session 58: Properties of hermitian matrices
session 59: Variational characterization of Eigenvalues: Rayleigh-Ritz theorem
session 60: Variational characterization of eigenvalues (continued)
session 61: Courant-Fischer theorem
session 62: Summary of Rayliegh-Ritz and Courant-Fischer theorems
session 63: Weyl's theorem
session 64: Positive semi-definite matrix, monotonicity theorem and interlacing theorems
session 65: Interlacing theorem I
session 66: Interlacing theorem II (Converse)
session 67: Interlacing theorem (Continued)
session 68: Eigenvalues: Majorization theorem and proof
session 69: Location and perturbation of Eigenvalues Part1: Dominant diagonal theorem
session 70: Location and perturbation of Eigenvalues Part2: Gersgorin's theorem
session 71: Implications of Gersgorin disc theorem, condition of eigenvalues
session 72: Condition of eigenvalues for diagonalizable matrices
session 73: Perturbation of eigenvalues Birkhoff's theorem Hoffman - Weilandt theorem
session 74: Singular value definition and some remarks.
session 75: Proof of singular value decomposition theorem.
session 76: Partitioning the SVD
session 77: Properties of SVD
session 78: Generalized inverse of matrices
session 79: Least squares
session 80: Constrained least squares

Catalogue records

nptel: 111108157 (manifestation)
nptel_video: 111108157 (manifestation)