Matrices for Engineers

Matrices for Engineers is designed to supply engineers and engineering students with a foundation in matrix theory and versatility in the manipulation of matrices. The book's approach provides the necessary material in a direct manner, with examples that illustrate each concept as it appears.The book emphasizes methodology and includes topics such as how to obtain the characteristic polynomial of a matrix; the factorizations of a coefficient matrix for ease of computation; and linear transformations from an intuitive and engineering point of view, in which conditions at one point in asystem induce conditions at another. Indeed, it covers computational techniques and goes beyond matrix algebra to include matrix calculus. Perfect for self-study, Matrices for Engineers also functions as a supplement to Oxford University Press's popular Linear Circuit Analysis, Second Edition (0-19-513666-7), by Raymond A. DeCarlo and Pen-Min Lin or any introductory electrical engineering text, such as Introduction to Electrical Engineering (0-19-513604-7) by Mulukutla S. Sarma. It can also be used to help in preparing for the Fundamentals of Engineering (FE)/Engineer-in-Training (EIT) exam and the Professional Engineer (PE) exam. For a complete and detailed list of engineering exam review books available from Oxford University Press, visit our website athttp://www.engineeringpress.com. Also Available from Oxford University Press DeCarlo and Lin's Linear Circuit Analysis, Second Edition (0-19-513666-7): Allan's Circuits Problems by Allan D. Kraus (0-19-514248-9) Solutions Manual to Accompany Linear Circuit Analysis, Second Edition, by Raymond A. DeCarlo and Pen-Min Lin (0-19-514218-7) Microsoft PowerPoint Overheads to Accompany Linear Circuit Analysis, Second Edition (0-19-514724-3) Sarma's Introduction to Electrical Engineering (0-19-513604-7):Solutions Manual to Accompany Introduction to Electrical Engineering by Mulukutla S. Sarma (0-19-514260-8) Microsoft PowerPoint Overheads to Accompany Introduction to Electrical Engineering (0-19-514472-4) KC's Problems and Solutions to Accompany Microelectronic Circuits, Fourth Edition, by K. C. Smith (0-19-511771-9) Spice, Second Edition, by Gordon Roberts and Adel Sedra (0-19-510842-6) Getting Started with MATLAB: A Quick Introduction for Scientists and Engineers (Version 6), by Rudra Pratap (0-19-515014-7)

Preface

xiii

Preliminary Concepts

1

(24)

Introduction

1

(1)

Basic Concepts

2

(1)

Matrix and Vector Terminology

3

(2)

Some Special Matrices

5

(2)

Matrix Equality

7

(1)

Matrix Addition and Substraction

7

(1)

Matrix Multiplication

8

(8)

Multiplication by a Scalar

8

(1)

The Scalar of Dot Product

9

(1)

Matrix Multiplication

10

(6)

Matrix Division and Matrix Inversion

16

(1)

Powers and Roots of Square Matrices

17

(1)

Further Properties of the Transpose

18

(2)

Some Special Matrices

20

(1)

Problems

21

(4)

Determinants

25

(26)

Introduction

25

(1)

Examples of the Solution of Simultaneous Equations

26

(3)

The Evaluation of Second and Third Order Determinants

29

(2)

Preliminary Rules for Operation with Determinants

31

(6)

Minors and Cofactors

37

(4)

The Minor

37

(1)

The Complementary Minor or Complement

38

(1)

The Algebraic Complement

39

(1)

The Cofactor

39

(2)

The Cofactor Matrix

41

(2)

Rules for Operation with Cofactors

43

(2)

Pivotal Condensation

45

(3)

Problems

48

(3)

Matrix Inversion

51

(30)

Introduction

51

(1)

Matrix Inversion

52

(1)

Properties of the Inverse

52

(2)

The Adjoint Matrix

54

(1)

The Determination of the Inverse

55

(5)

Cramer's Rule

60

(2)

Methods for Obtaining the Inverse

62

(1)

Elementary Transformation Matrices

63

(8)

The Inverse by Successive Transformations

71

(2)

The ``Sweep-Out'' Method

73

(2)

Labor Saving Possibilities

75

(3)

Problems

78

(3)

The Partitioning of Matrices

81

(24)

Introduction

81

(1)

Addition and Subtraction of Partitioned Matrices

82

(2)

Multiplication of Partitioned Matrices

84

(3)

The Transpose of a Partitioned Matrix

87

(1)

The Inverse of a Nonsymmetric Matrix by Partitioning

88

(5)

The Inverse of a Symmetric Matrix by Partitioning

93

(3)

Inverses of Complex Matrices

96

(6)

Preliminary Considerations

96

(1)

A and B Commute

97

(2)

A and B Do Not Commute

99

(3)

Problems

102

(3)

Simultaneous Equations

105

(36)

Introduction

105

(1)

Linear Dependence and Independence

106

(2)

Linear Combinations

108

(1)

The Rank of a Matrix

109

(1)

The Relationship Between Rank and Linear Dependence

110

(1)

Conditions for the Solution of Simultaneous Equations

111

(1)

Examples: Nonhomogeneous Cases

112

(3)

Homogeneous Equations

115

(1)

Equivalence of Matrices

116

(1)

Gaussian Elimination

117

(4)

Triangular Matrices

121

(4)

Triangular Decompositions

125

(1)

The Crout Reduction

126

(3)

Positive Definite Matrices

129

(2)

The Cholesky Reduction

131

(2)

The Forward and Back Substitution Algorithms

133

(4)

Problems

137

(4)

Orthogonality and Coordinate Transformations

141

(36)

Introduction

141

(1)

Orthogonal Vectors

141

(3)

The Scalar or Dot Product Revisited

144

(1)

Two Inequalities

144

(1)

The Angle Between Two Vectors

145

(2)

The Projection of One Vector on Another

147

(1)

Direction Cosines

148

(1)

The Linear Independence of a Set of Vectors

149

(3)

Orthogonal Matrices

152

(3)

The Gram-Schmidt Orthogonalization Process

155

(5)

Linear Transformations

160

(1)

The Geometric Interpretation of Linear Transformations

161

(3)

The Reflection Transformation

161

(1)

The Rotation Transformation

162

(1)

The Shear Transformation

163

(1)

The Transformation of Coordinates

164

(5)

Preliminary Considerations

164

(3)

Cross and Scalar Triple Products

167

(2)

The Transformations

169

(5)

Problems

174

(3)

Eigenvalues and Eigenvectors

177

(38)

Introduction

177

(2)

Eigenvalues

179

(1)

Properties of the Characteristic Polynomial

180

(1)

Methods for Finding p(λ)

181

(1)

The Method of Principal Minors

182

(1)

Bocher's Method

183

(2)

Leverrier's Method

185

(2)

Eigenvectors for Nonrepeated Eigenvalues

187

(4)

Eigenvectors for Repeated Eigenvalues

191

(3)

Eigenvectors for Complex Eigenvalues

194

(3)

The Modal Matrix

197

(1)

Further Considerations for Symmetric Matrices

198

(5)

Additional Properties

203

(3)

The Diagonal Form of a Matrix

206

(3)

Powers and Roots of Matrices

209

(3)

Problems

212

(3)

The Calculus of Matrices

215

(38)

Introduction

215

(1)

The Cayley-Hamilton Theorem

216

(8)

Introduction and Derivation

216

(3)

Two Applications

219

(5)

An Application of the Remainder Theorem

224

(1)

Infinite Series Involving Matrices

225

(1)

Sylvester's Theorem

226

(5)

Matrix Functions with Distinct Eigenvalues

231

(3)

Matrix Functions with Repeated Eigenvalues

234

(2)

Matrix Functions with Complex Eigenvalues

236

(2)

Another Method

238

(3)

The Solution of Differential Equations

241

(1)

The Integrating Factor and the Solution Method

242

(4)

Simultaneous Differential Equations

246

(3)

Problems

249

(4)

Index

253

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Matrices for Engineers

Summary

Table of Contents

Supplemental Materials