Matrix analysis for scientists & engineers / Alan J. Laub.

Format:

Book

Author/Creator:

Laub, Alan J., 1948-

Language:

English

Subjects (All):

Matrices.

Mathematical analysis.

Physical Description:

xiii, 157 pages : illustrations ; 26 cm

Other Title:

Matrix analysis for scientists and engineers

Place of Publication:

Philadelphia, PA : Society for Industrial and Applied Mathematics, [2005]

Summary:

Matrix Analysis for Scientists and Engineers provides a blend of undergraduate- and graduate-level topics in matrix theory and linear algebra that relieves instructors of the burden of reviewing such material in subsequent courses that depend heavily on the language of matrices. Consequently, the text provides an often-needed bridge between undergraduate-level matrix theory and linear algebra and the level of matrix analysis required for graduate-level study and research. The text is sufficiently compact that the material can be taught comfortably in a one-quarter or one-semester course.

Throughout the book, the author emphasizes the concept of matrix factorization to provide a foundation for a later course in numerical linear algebra. The author addresses connections to differential and difference equations as well as to linear system theory and encourages instructors to augment these examples with other applications of their own choosing.

Because the tools of matrix analysis are applied on a daily basis to problems in biology, chemistry, econometrics, engineering, physics, statistics, and a wide variety of other fields, the text can serve a rather diverse audience. The book is primarily intended to be used as a text for senior undergraduate or beginning graduate students in engineering, the sciences, mathematics, computer science, or computational science who wish to be familiar enough with matrix analysis and linear algebra that they can effectively use the tools and ideas of these fundamental subjects in a variety of applications. However, individual engineers and scientists who need a concise reference or a text for self-study will also find this book useful.

Prerequisites for using this text are knowledge of calculus and some previous exposure to matrices and linear algebra, including, for example, a basic knowledge of determinants, singularity of matrices, eigenvalues and eigenvectors, and positive definite matrices. There are exercises at the end of each chapter.

Contents:

1 Introduction and Review 1

1.1 Some Notation and Terminology 1

1.2 Matrix Arithmetic 3

1.3 Inner Products and Orthogonality 4

1.4 Determinants 4

2 Vector Spaces 7

2.2 Subspaces 9

2.3 Linear Independence 10

2.4 Sums and Intersections of Subspaces 13

3 Linear Transformations 17

3.1 Definition and Examples 17

3.2 Matrix Representation of Linear Transformations 18

3.3 Composition of Transformations 19

3.4 Structure of Linear Transformations 20

3.5 Four Fundamental Subspaces 22

4 Introduction to the Moore-Penrose Pseudoinverse 29

4.1 Definitions and Characterizations 29

4.3 Properties and Applications 31

5 Introduction to the Singular Value Decomposition 35

5.1 The Fundamental Theorem 35

5.2 Some Basic Properties 38

5.3 Row and Column Compressions 40

6 Linear Equations 43

6.1 Vector Linear Equations 43

6.2 Matrix Linear Equations 44

6.3 A More General Matrix Linear Equation 47

6.4 Some Useful and Interesting Inverses 47

7 Projections, Inner Product Spaces, and Norms 51

7.1 Projections 51

7.1.1 The four fundamental orthogonal projections 52

7.2 Inner Product Spaces 54

7.3 Vector Norms 57

7.4 Matrix Norms 59

8 Linear Least Squares Problems 65

8.1 The Linear Least Squares Problem 65

8.2 Geometric Solution 67

8.3 Linear Regression and Other Linear Least Squares Problems 67

8.3.1 Example: Linear regression 67

8.3.2 Other least squares problems 69

8.4 Least Squares and Singular Value Decomposition 70

8.5 Least Squares and QR Factorization 71

9 Eigenvalues and Eigenvectors 75

9.1 Fundamental Definitions and Properties 75

9.2 Jordan Canonical Form 82

9.3 Determination of the JCF 85

9.3.1 Theoretical computation 86

9.3.2 On the +1's in JCF blocks 88

9.4 Geometric Aspects of the JCF 89

9.5 The Matrix Sign Function 91

10 Canonical Forms 95

10.1 Some Basic Canonical Forms 95

10.2 Definite Matrices 99

10.3 Equivalence Transformations and Congruence 102

10.3.1 Block matrices and definiteness 104

10.4 Rational Canonical Form 104

11 Linear Differential and Difference Equations 109

11.1 Differential Equations 109

11.1.1 Properties of the matrix exponential 109

11.1.2 Homogeneous linear differential equations 112

11.1.3 Inhomogeneous linear differential equations 112

11.1.4 Linear matrix differential equations 113

11.1.5 Modal decompositions 114

11.1.6 Computation of the matrix exponential 114

11.2 Difference Equations 118

11.2.1 Homogeneous linear difference equations 118

11.2.2 Inhomogeneous linear difference equations 118

11.2.3 Computation of matrix powers 119

11.3 Higher-Order Equations 120

12 Generalized Eigenvalue Problems 125

12.1 The Generalized Eigenvalue/Eigenvector Problem 125

12.2 Canonical Forms 127

12.3 Application to the Computation of System Zeros 130

12.4 Symmetric Generalized Eigenvalue Problems 131

12.5 Simultaneous Diagonalization 133

12.5.1 Simultaneous diagonalization via SVD 133

12.6 Higher-Order Eigenvalue Problems 135

12.6.1 Conversion to first-order form 135

13 Kronecker Products 139

13.1 Definition and Examples 139

13.2 Properties of the Kronecker Product 140

13.3 Application to Sylvester and Lyapunov Equations 144.

Notes:

Includes bibliographical references (pages 151-152) and index.

ISBN:

0898715768

OCLC:

56686491