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Advanced Matrix Theory and Linear Algebra for Engineers

Advanced Matrix Theory and Linear Algebra for Engineers. Instructor: Prof. Vittal Rao, Centre for Electronics Design and Technology, IISc Bangalore. Introduction to systems of linear equations, Vector spaces, Solutions of linear systems, Important subspaces associated with a matrix, Orthogonality, Eigenvalues and eigenvectors, Diagonalizable matrices, Hermitian and symmetric matrices, General matrices. (from nptel.ac.in)

Lecture 28 - Diagonalization, Eigenvalues and Eigenvectors

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Prologue

Lecture 01 - Prologue Part 1: Systems of Linear Equations, Matrix Notation

Lecture 02 - Prologue Part 2: Diagonalization of a Square Matrix

Lecture 03 - Prologue Part 3: Homogeneous Systems, Elementary Row Operations

Linear Systems

Lecture 04 - Linear Systems 1: Elementary Row Operations (EROs)

Lecture 05 - Linear Systems 2: Row Reduced Echelon Form, The Reduction Process

Lecture 06 - Linear Systems 3: The Reduction Process, Solution using EROs

Lecture 07 - Linear Systems 4: Solution using EROs: Non-homogeneous Systems

Vector Spaces

Lecture 08 - Vector Spaces Part 1

Lecture 09 - Vector Spaces Part 2

Linear Independence and Subspaces

Lecture 10 - Linear Combination, Linear Independence and Dependence

Lecture 11 - Linear Independence and Dependence, Subspaces

Lecture 12 - Subspace Spanned by a Finite Set of Vectors, The Basic Subspaces Associated with a Matrix

Lecture 13 - Subspace Spanned by an Infinite Set of Vectors, Linear Independence of an Infinite Set of Vectors

Basis

Lecture 14 - Basis, Basis as a Maximal Linearly Independent Set

Lecture 15 - Finite Dimensional Vector Spaces

Lecture 16 - Extension of a Linearly Independent Set to a Basis, Ordered Basis

Linear Transformations

Lecture 17 - Relation between Representation in Two Bases, Linear Transformations

Lecture 18 - Examples of Linear Transformations

Lecture 19 - Null Space and Range of a Linear Transformation

Lecture 20 - Rank Nullity Theorem, One-One Linear Transformation

Lecture 21 - One-One Linear Transformation, Onto Linear Transformations, Isomorphisms

Inner Product and Orthogonality

Lecture 22 - Inner Product and Orthogonality

Lecture 23 - Orthonormal Sets, Orthonormal Basis and Fourier Expansion

Lecture 24 - Fourier Expansion, Gram-Schmidt Orthonormalization, Orthogonal Complements

Lecture 25 - Orthogonal Complements, Decomposition of a Vector, Pythagoras Theorem

Lecture 26 - Orthogonal Complements in the context of Subspaces Associated with a Matrix

Lecture 27 - Best Approximation

Diagonalization

Lecture 28 - Diagonalization, Eigenvalues and Eigenvectors

Lecture 29 - Eigenvalues and Eigenvectors, Characteristic Polynomial

Lecture 30 - Algebraic Multiplicity, Eigenvectors, Eigenspaces and Geometric Multiplicity

Lecture 31 - Criterion for Diagonalization

Hermitian and Symmetric Matrices

Lecture 32 - Hermitian and Symmetric Matrices, Unitary Matrix

Lecture 33 - Unitary and Orthogonal Matrices, Eigen Properties of Hermitian Matrices, Unitary Diagonalization

Lecture 34 - Spectral Decomposition

Lecture 35 - Positive and Negative Definite and Semidefinite Matrices

Singular Value Decomposition (SVD)

Lecture 36 - Singular Value Decomposition (SVD) Part 1

Lecture 37 - Singular Value Decomposition (SVD) Part 2

Back to Linear Systems

Lecture 38 - Back to Linear Systems Part 1

Lecture 39 - Back to Linear Systems Part 2

Epilogue

Lecture 40 - Epilogue