Fundamentals of matrix analysis with applications / by Edward Barry Saff,; Arthur David Snider

By: Saff, E. B, 1944-Contributor(s): Snider, Arthur David, 1940-Material type: TextTextPublication details: Delhi: Wiley , 2018cDescription: xi, 395p.: ill.; 24 cmISBN: 9788126574186 (pbk)Subject(s): Matrices | Algebras, Linear | Orthogonalization methods | EigenvaluesGenre/Form: Electronic books.DDC classification: 512.9434 Online resources: Full text available from Ebook Central - Academic Complete
Contents:
Title Page -- Copyright Page -- Contents -- Preface -- PART I INTRODUCTION: THREE EXAMPLES -- Chapter 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 -- Chapter 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 -- Group Projects for Part I -- A. LU Factorization -- B. Two-Point Boundary Value Problem -- C. Electrostatic Voltage -- D. Kirchhoff's Laws -- E. Global Positioning Systems -- F. Fixed-Point Methods -- PART II INTRODUCTION: THE STRUCTURE OF GENERAL SOLUTIONS TO LINEAR ALGEBRAIC EQUATIONS -- Chapter 3 Vector Spaces -- 3.1 General Spaces, Subspaces, and Spans -- 3.2 Linear Dependence -- 3.3 Bases, Dimension, and Rank -- 3.4 Summary -- Chapter 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 -- Group Projects for Part II -- A. Rotations and Reflections -- B. Householder Reflectors -- C. Infinite Dimensional Matrices -- PART III INTRODUCTION: REFLECT ON THIS -- Chapter 5 Eigenvectors and Eigenvalues -- 5.1 Eigenvector Basics -- 5.2 Calculating Eigenvalues and Eigenvectors -- 5.3 Symmetric and Hermitian Matrices -- 5.4 Summary -- Chapter 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 -- Chapter 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 III -- 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 -- EULA.
Summary: This book provides comprehensive coverage of matrix theory from a geometric and physical perspective, and the authors address the functionality of matrices and their ability to illustrate and aid in many practical applications. Readers are introduced to inverses and eigenvalues through physical examples such as rotations, reflections, and projections, and only then are computational details described and explored. MATLAB is utilized to aid in reader comprehension, and the authors are careful to address the issue of rank fragility so readers are not flummoxed when MATLAB displays conflict with manual calculations. Matrix foundations are exphasized via projects involving LU factorizations and the matrix aspects of finite difference modeling and Kirchhoff's circuit laws. Vector space concepts and the many facets of orthogonality are then discussed, and in an effort maintain a computational perpective, attention is directed to the numerical issues of error control through norm preservation. Projects include rotational kinematics, Householder implementation of QR factorizations, and the infinite dimensional matrices arising in Haar wavelet formulations. The statistical unlikeliness of singular square matrices, multiple eignevalues, and defective matrices are then emphasized for random matrices, and the basic workings of the QR algorithm (and the role of luck in its implementation as well as in the occurrence of defective matrices) and the random-shift amelioration of its failures are explored. The book concludes with a chapter on the role of matrices in the solution of linear systems of diffential equations (DEs) with constant coefficients via the matrix exponential. Insight into the ssues related to its computation are also provided.
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Holdings
Item type Current library Call number Copy number Status Date due Barcode Item holds
Books Books Namal Library
Mathematics 		512.9434 SAF-F 2018 10953 (Browse shelf (Opens below)) 1 Available 0010953
Books Books Namal Library
Mathematics 		512.9434 SAF-F 2018 10954 (Browse shelf (Opens below)) 2 Available 0010954
Total holds: 0

Includes bibliographical references and index.

Title Page --
Copyright Page --
Contents --
Preface --
PART I INTRODUCTION: THREE EXAMPLES --
Chapter 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 --
Chapter 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 --
Group Projects for Part I --
A. LU Factorization --
B. Two-Point Boundary Value Problem --
C. Electrostatic Voltage --
D. Kirchhoff's Laws --
E. Global Positioning Systems --
F. Fixed-Point Methods --
PART II INTRODUCTION: THE STRUCTURE OF GENERAL SOLUTIONS TO LINEAR ALGEBRAIC EQUATIONS --
Chapter 3 Vector Spaces --
3.1 General Spaces, Subspaces, and Spans --
3.2 Linear Dependence --
3.3 Bases, Dimension, and Rank --
3.4 Summary --
Chapter 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 --
Group Projects for Part II --
A. Rotations and Reflections --
B. Householder Reflectors --
C. Infinite Dimensional Matrices --
PART III INTRODUCTION: REFLECT ON THIS --
Chapter 5 Eigenvectors and Eigenvalues --
5.1 Eigenvector Basics --
5.2 Calculating Eigenvalues and Eigenvectors --
5.3 Symmetric and Hermitian Matrices --
5.4 Summary --
Chapter 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 --
Chapter 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 III --
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 --
EULA.


This book provides comprehensive coverage of matrix theory from a geometric and physical perspective, and the authors address the functionality of matrices and their ability to illustrate and aid in many practical applications. Readers are introduced to inverses and eigenvalues through physical examples such as rotations, reflections, and projections, and only then are computational details described and explored. MATLAB is utilized to aid in reader comprehension, and the authors are careful to address the issue of rank fragility so readers are not flummoxed when MATLAB displays conflict with manual calculations. Matrix foundations are exphasized via projects involving LU factorizations and the matrix aspects of finite difference modeling and Kirchhoff's circuit laws. Vector space concepts and the many facets of orthogonality are then discussed, and in an effort maintain a computational perpective, attention is directed to the numerical issues of error control through norm preservation. Projects include rotational kinematics, Householder implementation of QR factorizations, and the infinite dimensional matrices arising in Haar wavelet formulations. The statistical unlikeliness of singular square matrices, multiple eignevalues, and defective matrices are then emphasized for random matrices, and the basic workings of the QR algorithm (and the role of luck in its implementation as well as in the occurrence of defective matrices) and the random-shift amelioration of its failures are explored. The book concludes with a chapter on the role of matrices in the solution of linear systems of diffential equations (DEs) with constant coefficients via the matrix exponential. Insight into the ssues related to its computation are also provided.

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