Linear Algebra, Matrix Theory and Applications

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. 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 Provides insights into the various aspects related to matrices, including the concepts of vector spaces, least square regression, determinants, eigen values, eigen vectors, positive definite matrices, and singular value decomposition, and explains the methods of computation in matrices.