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Full Description
This set includes Fundamentals of Matrix Analysis with Applications & Solutions Manual to Accompany Fundamentals of Matrix Analysis with Applications
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 precision, complex arithmetic formulations, parameterization of solutions, and the logical traps that dictate strict adherence to Gauss's instructions. The book heralds matrix formulation both as notational shorthand and as a quantifier of physical operations such as rotations, projections, reflections, and the Gauss reductions. Inverses and eigenvectors are visualized first in an operator context before being addressed computationally. Least squares theory is expounded in all its manifestations including optimization, orthogonality, computational accuracy, and even function theory. Fundamentals of Matrix Analysis with Applications also features:
Novel approaches employed to explicate the QR, singular value, Schur, and Jordan decompositions and their applications
Coverage of the role of the matrix exponential in the solution of linear systems of differential equations with constant coefficients
Chapter-by-chapter summaries, review problems, technical writing exercises, select solutions, and group projects to aid comprehension of the presented concepts
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
Chapter 2. MATRIX ALGEBRA
2.1 Matrix Multiplication
2.2 Some Applications of Matrix Operators
2.3 The Inverse and the Transpose
2.4 Determinants
2.5 Three Important Determinant Rules
Review Problems for Part I
Technical Writing Exercises for Part I
Group Projects for Part I
A. LU Factorization
B. Two]Point Boundary Value Problems
C. Electrostatic Voltage
D. Kirchhoff's Laws
E. Global Positioning Systems
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
Chapter 4. ORTHOGONALITY
4.1 Orthogonal Vectors and the Gram]Schmidt Algorithm Norm
4.2 Orthogonal Matrices
4.3 Least Squares
4.4 Function Spaces
Review Problems for Part II
Magic square
Controllability
Technical Writing Exercises for Part II
Group Projects for Part II
A. Orthogonal Matrices, Rotations, and Reflections
B. Householder Reflectors and the QR Factorization
C. Infinite Dimensional Matrices
Part III
Introduction: Reflect on This
Chapter 5. Eigenvalues and Eigenvectors
5.1 Eigenvector Basics
5.2 Calculating Eigenvalues and Eigenvectors
5.3 Symmetric and Hermitian Matrices
Chapter 5. Summary
Chapter 6. Similarity
6.1 Similarity Transformations and Diagonalizability
6.2 Principal Axes Normal Modes
6.3 Schur Decomposition and Its Implications
6.4 The Power Method and the QR Algorithm
Chapter 7. Linear Systems of Differential Equations
7.1 First Order Linear Systems of Differential Equations
7.2 The Matrix Exponential Function
7.3 The Jordan Normal Form
Review Problems for Part III
Technical Writing Exercises for Part III
Group Projects for Part III
A. Positive Definite Matrices
B. Hessenberg Form
C. The Discrete Fourier Transform and Circulant Matrices
Answers to Odd]Numbered Problems
Index
