The study of matrices and their applications forms a fundamental part of modern mathematics and is crucial across numerous disciplines such as physics, computer science, data science, engineering, statistics and economics. The purpose of this book is to provide a clear, systematic and comprehensive understanding of matrix theory and its practical relevance, especially for undergraduate and postgraduate students in mathematical and applied sciences.
This book is carefully structured to begin with the foundational concepts of matrices, their types, and operations, gradually progressing to more advanced topics like rank, solutions of linear systems, eigen values and eigen vectors. We also cover matrix decompositions — LU and QR — which are essential tools in numerical linear algebra and optimization, particularly useful in machine learning and scientific computing.
Each chapter is enriched with solved examples, step-by-step procedures, and “Do Yourself” exercises designed to reinforce understanding and build confidence. Special attention has been given to pedagogical clarity so that even students with minimal prior exposure to linear algebra can grasp the subject with ease. The inclusion of Python code snippets further bridges the gap between theoretical concepts and computational implementation, fostering algorithmic thinking among learners.
This book also serves as a valuable reference for educators and researchers, offering insights into how matrix theory connects to practical problem-solving in real-world contexts.
Contents –
Chapter 1 Matrices
1.1 Introduction to Matrix
1.2 Definition
1.3 Types of Matrices
1.4 Properties of Transpose of Matrix
1.5 Linear Independent Row and Linear Independent Columns
1.6 Linear Dependent Row and Linear Dependent Columns
1.7 Rank of Matrix
1.8 Solution of a System of Linear Equations by Rank Method
Chapter 2 Eigen Values and Eigen Vectors
2.1 Introduction to Eigen Values and Eigen Vectors
2.2 Definition
2.3 Characteristic Polynomial
2.4 Characteristic Equation
2.5 Properties of Eigen Values
2.6 Spectrum of Eigen Values of Various Matrices
2.7 Properties of Eigen Vectors
2.8 Some Important Facts of the Matrix
Chapter 3 LU Decomposition
3.1 Introduction
3.2 Definition
3.3 LU Decomposition by Row Operation Method
3.4 General Python Code for LU Decomposition
3.5 Python Algorithm Using Row Operations for LU Decomposition
Chapter 4 QR Decomposition
4.1 Introduction
4.2 Definition
4.3 QR Decomposition is Useful Why?
4.4 Gram Schmidt Method for QR Decomposition
4.5 Application of QR decomposition in Machine Learning
Chapter 5 Singular Value Decomposition (SVD), PCA, and Applications
5.1 Introduction to Singular Value Decomposition (SVD)
5.2 Definition
5.3 Intuition behind SVD
5.4 SVD of Matrix A
5.5 Application in Image Compression
5.6 Introduction to Machine Learning (ML)
5.7 ML Types
5.8 Principal Component Analysis (PCA)
5.9 How to Implement PCA?
5.10 Matrix and Machine Learning Concept in Indian
Knowledge System