# Fundamentals of Matrix Analysis with Applications

ISBN: 9788126574186

408 pages

Exclusively distributed by I.K. International

## Description

Providing comprehensive coverage of matrix theory from a geometric and physical perspective, Fundamentals of Matrix Analysis with Applications describes the functionality of matrices and their ability to quantify and analyze many practical applications. Written by a highly-qualified author team, the book presents tools for matrix analysis and is illustrated with extensive examples and software implementations. Beginning with a detailed exposition and review of the Gauss elimination method, the authors maintain readers' interest with refreshing discussions regarding the issues of operation counts, computer speed and prevision, complex arithmetic formulations, parametrization of solutions, and the logical traps that dictate strict adherence to Gauss's instructions.

Preface

Part I Introduction: Three Examples

1 Systems of Linear Algebraic Equations

1.1 Linear Algebraic Equations

1.2 Matrix Representation of Linear Systems and the Gauss-Jordan Algorithm

1.3 The Complete Gauss Elimination Algorithm

1.4 Echelon Form and Rank

1.5 Computational Considerations

1.6 Summary

2 Matrix Algebra

2.1 Matrix Multiplication

2.2 Some Physical Applications of Matrix Operators

2.3 The Inverse and the Transpose

2.4 Determinants

2.5 Three Important Determinant Rules

2.6 Summary

Part II Introduction: The Structure of General Solutions to Linear Algebraic Equations

3 Vector Spaces

3.1 General Spaces Subspaces and Spans

3.2 Linear Dependence

3.3 Bases Dimension and Rank

3.4 Summary

4 Orthogonality

4.1 Orthogonal Vectors and the Gram--Schmidt Algorithm

4.2 Orthogonal Matrices

4.3 Least Squares

4.4 Function Spaces

4.5 Summary

Part III Introduction: Reflect on This

5 Eigenvectors and Eigenvalues

5.1 Eigenvector Basics

5.2 Calculating Eigenvalues and Eigenvectors

5.3 Symmetric and Hermitian Matrices

5.4 Summary

6 Similarity

6.1 Similarity Transformations and Diagonalizability

6.2 Principle Axes and Normal Modes

6.3 Schur Decomposition and Its Implications

6.4 The Singular Value Decomposition

6.5 The Power Method and the QR Algorithm

6.6 Summary

7 Linear Systems of Differential Equations

7.1 First-Order Linear Systems

7.2 The Matrix Exponential Function

7.3 The Jordan Normal Form

7.4 Matrix Exponentiation via Generalized Eigenvectors

7.5 Summary

Group Projects for Part

A. Positive Definite Matrices

B. Hessenberg Form

C. Discrete Fourier Transform

D. Construction of the SVD

E. Total Least Squares

F. Fibonacci Numbers

Answers to Odd Numbered Exercises

Index