Applied and Computational Linear Algebra: A First Course

[Pages:504]Charles L. Byrne Department of Mathematical Sciences University of Massachusetts Lowell

Applied and Computational Linear Algebra: A First Course

To Eileen, my wife for the last forty-three years.

My thanks to David Einstein, who read most of an earlier version of this book

and made many helpful suggestions.

Contents

Preface

xxiii

I Preliminaries

1

1 Introduction

1

1.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of this Course . . . . . . . . . . . . . . . . . . . . 1 1.3 Solving Systems of Linear Equations . . . . . . . . . . . . . 2 1.4 Imposing Constraints . . . . . . . . . . . . . . . . . . . . . 2 1.5 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.6 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 An Overview of Applications

5

2.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Transmission Tomography . . . . . . . . . . . . . . . . . . 6

2.2.1 Brief Description . . . . . . . . . . . . . . . . . . . . 6 2.2.2 The Theoretical Problem . . . . . . . . . . . . . . . 7 2.2.3 The Practical Problem . . . . . . . . . . . . . . . . . 7 2.2.4 The Discretized Problem . . . . . . . . . . . . . . . 8 2.2.5 Mathematical Tools . . . . . . . . . . . . . . . . . . 8 2.3 Emission Tomography . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Coincidence-Detection PET . . . . . . . . . . . . . . 9 2.3.2 Single-Photon Emission Tomography . . . . . . . . . 9 2.3.3 The Line-Integral Model for PET and SPECT . . . 10 2.3.4 Problems with the Line-Integral Model . . . . . . . . 10 2.3.5 The Stochastic Model: Discrete Poisson Emitters . . 11 2.3.6 Reconstruction as Parameter Estimation . . . . . . . 11 2.3.7 X-Ray Fluorescence Computed Tomography . . . . . 12 2.4 Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . 12 2.4.1 Alignment . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.2 Precession . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.3 Slice Isolation . . . . . . . . . . . . . . . . . . . . . . 13 2.4.4 Tipping . . . . . . . . . . . . . . . . . . . . . . . . . 13

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Contents

2.4.5 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.6 The Line-Integral Approach . . . . . . . . . . . . . . 14 2.4.7 Phase Encoding . . . . . . . . . . . . . . . . . . . . 14 2.4.8 A New Application . . . . . . . . . . . . . . . . . . . 14 2.5 Intensity Modulated Radiation Therapy . . . . . . . . . . . 15 2.5.1 Brief Description . . . . . . . . . . . . . . . . . . . . 15 2.5.2 The Problem and the Constraints . . . . . . . . . . 15 2.5.3 Convex Feasibility and IMRT . . . . . . . . . . . . . 15 2.6 Array Processing . . . . . . . . . . . . . . . . . . . . . . . . 16 2.7 A Word about Prior Information . . . . . . . . . . . . . . . 17

3 A Little Matrix Theory

21

3.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Matrix Operations . . . . . . . . . . . . . . . . . . . 24 3.3.2 Matrix Inverses . . . . . . . . . . . . . . . . . . . . . 25 3.3.3 The Sherman-Morrison-Woodbury Identity . . . . . 26 3.4 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . 27 3.4.1 Linear Independence and Bases . . . . . . . . . . . . 27 3.4.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4.3 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . 30 3.5 Representing a Linear Transformation . . . . . . . . . . . . 31 3.6 The Geometry of Euclidean Space . . . . . . . . . . . . . . 32 3.6.1 Dot Products . . . . . . . . . . . . . . . . . . . . . . 32 3.6.2 Cauchy's Inequality . . . . . . . . . . . . . . . . . . 34 3.7 Vectorization of a Matrix . . . . . . . . . . . . . . . . . . . 34 3.8 Solving Systems of Linear Equations . . . . . . . . . . . . . 35 3.8.1 Row-Reduction . . . . . . . . . . . . . . . . . . . . . 35 3.8.2 Row Operations as Matrix Multiplications . . . . . . 37 3.8.3 Determinants . . . . . . . . . . . . . . . . . . . . . . 37 3.8.4 Sylvester's Nullity Theorem . . . . . . . . . . . . . . 38 3.8.5 Homogeneous Systems of Linear Equations . . . . . 39 3.8.6 Real and Complex Systems of Linear Equations . . . 41 3.9 Under-Determined Systems of Linear Equations . . . . . . 41 3.10 Over-Determined Systems of Linear Equations . . . . . . . 43

4 The ART, MART and EM-MART

45

4.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 The ART in Tomography . . . . . . . . . . . . . . . . . . . 46 4.4 The ART in the General Case . . . . . . . . . . . . . . . . 47

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