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# Fundamentals of Matrix Analysis with Applications Set

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This set includes

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