Course Material for EE2100: Matrix Theory (Fall 2023)

Please Note that the course material is typically updated after each lecture. The topics listed for future dates are tentative and can vary.

Lecture Notes and Classroom Scribble

The classroom scribble and the lecture notes are intended for quick reference\review. Lecture notes and classroom scribble are by no means a complete source and infact are based on the suggested textbooks\references. It is highly recommended that the students also refer to the suggested textbooks\references.

(31 Jul 2023): Introduction [Classroom Scribble]
1. Course overview
2. Applications of Matrices
(02 Aug 2023): Vectors and Elementary Operations on Vectors [Classroom Scribble] [Lecture Notes]
1. Vectors and their properties
2. Elementary operations on Vectors: Vector Addition, Scalar Multiplication
3. Dot Product
(03 Aug 2023): Norm and Cauchy-Schwarz inequality [Classroom Scribble] [Lecture Notes]
1. Geometrical Interpretation of a Vector and Dot Product
2. Norm and its Geometrical Interpretation
3. Cauchy-Schwarz Inequality
(07 Aug 2023): Projection [Classroom Scribble] [Lecture Notes]
1. Angle between vectors
2. Projection
(09 Aug 2023): Vector Space [Classroom Scribble] [Lecture Notes]
1. Binary Operation and Field
2. Vector Space
(10 Aug 2023): Subspace and Span [Classroom Scribble] [Lecture Notes]
1. Few identities of Vector Space
2. Supspace of a Vector Space
3. Linear combination and Span
(14 Aug 2023): Linear Independence, Spanning Set and Basis [Classroom Scribble] [Lecture Notes]
1. Span as a subspace
2. Linear independence
3. Spanning set and Basis
(16 Aug 2023): Dimension of a Subspace [Classroom Scribble] [Lecture Notes]
1. Dimension of a Subspace
2. Theorems related to dimension of a Subspace
(17 Aug 2023): Orthogonal Basis [Classroom Scribble] [Lecture Notes]
1. Orthogonal Basis
2. Coordinate vector
3. Representation of a vector in Orthogonal Basis
(21 Aug 2023): Generation of Orthogonal Basis [Classroom Scribble] [Lecture Notes]
1. Grahm Schmidt Algorithm
2. Projection of a Vector onto a Subspace
(23 Aug 2023): Projection of a Vector onto a Subspace [Classroom Scribble] [Lecture Notes]
1. Projection of a vector onto a Subspace
2. Introduction to k-Means clustering
(24 Aug 2023): k-Means clustering [Classroom Scribble]
1. k-Means clustering
2. Convergence Aspects
(28 Aug 2023): Introduction to Matrices [Classroom Scribble] [Lecture Notes]
1. Introduction to Matrices
2. Elementary Matrix Operations
(30 Aug 2023): Matrix Vector Product [Classroom Scribble] [Lecture Notes]
1. Matrix Vector Product
2. Introduction to Linear Transformation
(31 Aug 2023): Linear Transformation and Matrices [Classroom Scribble] [Lecture Notes]
1. Linear Transformations and Transformation Matrices
2. Transformation matrices in 2D ( $mathcal{R}^2rightarrowmathcal{R}^2$ )
(04 Sep 2023): Inverse of a Matrix [Classroom Scribble] [Lecture Notes]
1. Matrix Multiplication
2. Elementary Row operations and Matrix Multiplication
(06 Sep 2023): Matrix Multiplication [Classroom Scribble] [Lecture Notes]
1. Matrix Vector Product
2. Introduction to Linear Transformation
(07 Aug 2023): Fundamental Subspaces [Classroom Scribble] [Lecture Notes]
1. System of Linear Equations
2. Fundamental Subspaces of a Matrix
(11 Sep 2023): Rank-Nullity Theorem [Classroom Scribble] [Lecture Notes]
1. Rank and Nullity of a Matrix
2. Rank-Nullity Theorem
(13 Sep 2023): Gaussian Elimination [Classroom Scribble] [Lecture Notes]
1. Back-Substitution to solve
2. Forward-Elimination
(14 Sep 2023): Gaussian Elimination [Classroom Scribble] [Lecture Notes]
1. Gaussian Elimination for Characterizing Null Space
2. Inconsistent system of linear equations
(18 Sep 2023): LU Decomposition [Classroom Scribble]
1. Introduction to LU decomposition
2. Elementary row operations and lower triangular matrices
(20 Sep 2023): LU Decomposition [Classroom Scribble]
1. Inverse of lower triangular matrices corresponding to elementary row operations
2. LU decomposition and solving system of linear equations
(21 Sep 2023): Overdetermined System of linear equations [Classroom Scribble] [Lecture Notes]
1. Overdetermined system of linear equations
2. Projection Matrix
(25 Sep 2023): Linear Regression [Lecture Notes]
1. Overview of Regression
2. Linear Regression
(04 Oct 2023): Trace and Determinant [Classroom Scribble] [Lecture Notes]
1. Trace of a Matrix
2. Introduction to Determinant of a Matrix
(05 Oct 2023): Determinant of a Matrix [Lecture Notes]
1. Determinant of a Matrix
2. Computation of Determinant
(09 Oct 2023): Eigen Values and Eigen Vectors [Lecture Notes]
1. Introduction to Eigen Values and Eigen Vectors
(11 Oct 2023): Eigen Values and Eigen Vectors [Lecture Notes]
1. Eigen Values and Eigen Vectors of Real and Symmetric Matrices
2. LU decomposition and solving system of linear equations
(12 Oct 2023): Spectral Theorem [Lecture Notes]
1. Quadratic Forms
2. Rayleigh Quotient
(16 Oct 2023): Positive Definite Matrices [Classroom Scribble] [Lecture Notes]
1. Quadratic Forms
2. Positive and Negative definite/semidefinite matrices
(18 Oct 2023): QR Decomposition [Classroom Scribble] [Lecture Notes]
1. Grahm Schmidt and QR Decomposition
2. Householder Transformation
(19 Oct 2023): Cholesky Decomposition [Classroom Scribble]
1. Decomposition of Positive semidefinite Matrices
2. Cholesky Decomposition
(30 Oct 2023): Decompositions and Solving Systems of Linear Equations
1. Cholesky Decomposition and Solving System of Linear Equations
2. Dolittle Method for LU Decomposition
(01 Nov 2023): Singular Value Decomposition [Classroom Scribble] [Lecture Notes]
1. Introduction to Singular Value Decomposition
2. Singular Values and Singular Vectors
(02 Nov 2023): Norm of a Matrix [Classroom Scribble] [Lecture Notes]
1. Representation of Matrix as a Sum of Rank-1 Matrices
2. Norm of a matrix
(06 Nov 2023): Condition Number of a Matrix [Classroom Scribble] [Lecture Notes]
1. Significance of Condition Number
2. Condition Number of a Matrix
(08 Nov 2023): Principal Component Analysis
1. Need for dimensionality reduction
2. Mean and Variance of Data sets.
(09 Nov 2023): Principal Component Analysis
1. Role of Singular Values in Principal Component Analysis
2. Algorithm for dimensionality reduction
(15 Nov 2023): Jordan Forms
1. Defective Matrices
2. Generalized Eigen Values
3. Jordan Forms
(20 Nov 2023): Similar Matrices
1. Notion of Similarity between Matrices.
2. Properties of Similar Matrices.