Linear Algebra, Matrix Theory and Applications

✍ Scribed by Stefano Spezia (editor)

• Publisher: Arcler Press
• Year: 2019
• Tongue: English
• Leaves: 344
• Category: Library

✦ Synopsis

Linear Algebra, Matrix Theory and Applications gives insights into the various aspects related to the matrices including the concepts on vector spaces, least square regression, determinants, eigen values, eigen vectors, positive definite matrices, singular value decomposition and teaches the readers the methods of computation in matrices.This book also discusses about Reduced triangular form of polynomial, Gaussian Elimination-based correlation analysis, Shift-invert diagonalization of spin chains, Fast matrix multiplication, Finding the pth root of principal matrix and Quasi-rational canonical form of a matrix.

✦ Table of Contents

Cover
Title Page
Copyright
DECLARATION
ABOUT THE EDITOR
TABLE OF CONTENTS
List of Contributors
List of Abbreviations
Preface
SECTION 1: MATRICES AND GAUSSIAN ELIMINATION
Chapter 1 Reduced Triangular Form of Polynomial 3-by-3 Matrices with One Characteristic Root and Its Invariants
Abstract
Introduction
Preliminary Results
Improvement of the Triangular Form of Matrix In the Class of Semiscalarly Equivalent Matrix: Reduced Matrix
Invariants of The Reduced Matrix
References
Chapter 2 Representation of the Matrix for Conversion between Triangular Bezier Patches and Rectangular Bezier Patches
Abstract
Construction of the Conversion Matrices
Conclusion
References
Chapter 3 Gaussian Elimination-Based Novel Canonical Correlation Analysis Method for EEG Motion Artifact Removal
Abstract
Introduction
Artifact Removal Methods
Proposed Algorithm
Eeg Signal Data Set
Result and Discussion
Conclusion
References
SECTION 2: VECTOR SPACES, LEAST SQUARES REGRESSION AND GRAM–SCHMIDT PROCESS
Chapter 4 Dimensional Lifting Through The Generalized Gram–Schmidt Process
Abstract
References
SECTION 3: DETERMINANTS
Chapter 5 On the Extension of Sarrus’ Rule to n × n (n > 3) Matrices: Development of New Method for the Computation of the Determinant of 4×4 Matrix
Abstract
Introduction
Definition of Determinants
Existing Methods of Computation of Determinants
The Development of the New Methods for the Computation of Determinants
Numerical Examples
Efficiency of The New Method
Programming
Conclusion and Future Works
References
Chapter 6 Optimization of the Determinant of the Vandermonde Matrix and Related Matrices
Abstract
Introduction
The Vandermonde Matrix
Application To D-Optimal Experiment Designs For Polynomial Regression With A Cost-Function
Optimization Using Grobner Bases
Extreme Points on The Ellipsoid In Three Dimensions
Extreme Points on The Cylinder In Three Dimensions
Optimizing the Vandermonde Determinant on a Surface Defined by a Homogeneous Polynomial
The Vandermonde Determinant on P-Norm Spheres
Conclusion
References
SECTION 4: EIGENVALUES AND EIGENVECTORS
Chapter 7 On Finite Nilpotent Matrix Groups Over Integral Domains
Abstract
Introduction
Acknowledgment
References
Chapter 8 A New Approach for Computing the Solution of Sylvester Matrix Equation
Abstract
Introduction
The Solution of Sylvester Matrix Equation
Numerical Experiments and Applications
Conclusion
Acknowledgments
References
Chapter 9 Shift-invert Diagonalization of Large Many-body Localizing Spin Chains
Abstract
Introduction
Description of The Problem
The Shift-Invert Technique
Benchmarks and Optimal Use of The Shift-Invert Method For The Mbl Problem
Reliability of Single Precision Results
Results For Large Systems
Discussion and Conclusion
Acknowledgements
A Shift-Invert Example Code
References
SECTION 5: POSITIVE DEFINITE MATRICES AND SINGULAR VALUE DECOMPOSITION
Chapter 10 Ordering Positive Definite Matrices
Abstract
Introduction
Homogeneous Geometry of Sn+
Affine-Invariant Orders
Monotone Functions on Sn+
Invariant Half-Spaces
Matrix Means
Conclusion
Acknowledgements
References
Chapter 11 Split-and-Combine Singular Value Decomposition for Large-Scale Matrix
Abstract
Introduction
Methodology
Svd For Continuously Growing Data
Experimental Result
Conclusion
Acknowledgment
References
SECTION 6: COMPUTATIONS WITH MATRICES AND LINEAR PROGRAMMING
Chapter 12 Fast Matrix Multiplication
Abstract
Introduction
Computations And Costs
Evaluation Of Polynomials
Bilinear Problems
The Exponent Of Matrix Multiplication
Border Rank
Schönhage’s .-Theorem
Strassen’s Laser Method
Coppersmith and Winograd’s Method
Group-Theoretic Approach
Applications
Support Rank
References
SECTION 7: THE JORDAN FORM AND THE PRINCIPAL MATRIX pTH ROOT
Chapter 13 Quasi-Rational Canonical Forms of a Matrix Over a Number Field
Abstract
Introduction
Jordan and Rational Canonical Forms
The Elementary Divisors of a Matrix Over a Number Field
Quasi-Rational Form of a Matrix
Conclusion
Acknowledgements
References
Index
Back Cover