Linear Algebra and Linear Operators in Engineering (eBook)

With Applications in Mathematica(R)

H. Ted Davis, Kendall T. Thomson (Autoren)

eBook Download: PDF | EPUB

2000 | 1. Auflage
547 Seiten
Elsevier Science (Verlag)
978-0-08-051024-8 (ISBN)

Infos

Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical sciences. Also included are several numerical applications, complete with Mathematica solutions and code, giving the student a hands-on introduction to numerical analysis. Linear Algebra and Linear Operators in Engineering is ideally suited as the main text of an introductory graduate course, and is a fine instrument for self-study or as a general reference for those applying mathematics.

? Contains numerous Mathematica examples complete with full code and solutions
? Provides complete numerical algorithms for solving linear and nonlinear problems
? Spans elementary notions to the functional theory of linear integral and differential equations
? Includes over 130 examples, illustrations, and exercises and over 220 problems ranging from basic concepts to challenging applications
? Presents real-life applications from chemical, mechanical, and electrical engineering and the physical sciences

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Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical sciences. Also included are several numerical applications, complete with Mathematica solutions and code, giving the student a "e;hands-on"e; introduction to numerical analysis. Linear Algebra and Linear Operators in Engineering is ideally suited as the main text of an introductory graduate course, and is a fine instrument for self-study or as a general reference for those applying mathematics. Contains numerous Mathematica examples complete with full code and solutions Provides complete numerical algorithms for solving linear and nonlinear problems Spans elementary notions to the functional theory of linear integral and differential equations Includes over 130 examples, illustrations, and exercises and over 220 problems ranging from basic concepts to challenging applications Presents real-life applications from chemical, mechanical, and electrical engineering and the physical sciences

Inhalt

Cover 1
Linear Algebra and Linear Operators in Engineering 4
Copyright Page 5
Contents 6
Preface 12
Chapter 1. Determinants 14
1.1. Synopsis 14
1.2. Matrices 15
1.3. Definition of a Determinant 16
1.4. Elementary Properties of Determinants 19
1.5. Cofactor Expansions 22
1.6. Cramer's Rule for Linear Equations 27
1.7. Minors and Rank of Matrices 29
Problems 31
Further Reading 35
Chapter 2. Vectors and Matrices 38
2.1. Synopsis 38
2.2. Addition and Multiplication 39
2.3. The Inverse Matrix 41
2.4. Transpose and Adjoint 46
2.5. Partitioning Matrices 48
2.6. Linear Vector Spaces 51
Problems 56
Further Reading 59
Chapter 3. Solution of Linear and Nonlinear Systems 60
3.1. Synopsis 60
3.2. Simple Gauss Elimination 61
3.3. Gauss Elimination with Pivoting 68
3.4. Computing the Inverse of a Matrix 71
3.5. LU-Decomposition 74
3.6. Band Matrices 79
3.7. Iterative Methods for Solving Ax = b 91
3.8. Nonhnear Equations 98
Problems 121
Further Reading 134
Chapter 4. General Theory of Solvability of Linear Algebraic Equations 136
4.1. Synopsis 136
4.2. Sylvester's Theorem and the Determinants of Matrix Products 137
4.3. Gauss-Jordan Transformation of a Matrix 142
4.4. General Solvability Theorem for Ax = b 146
4.5. Linear Dependence of a Vector Set and the Rank of Its Matrix 163
4.6. The Fredholm Alternative Theorem 168
Problems 172
Further Reading 174
Chapter 5. The Eigenproblem 176
5.1. Synopsis 176
5.2. Linear Operators in a Normed Linear Vector Space 178
5.3. Basis Sets in a Normed Linear Vector Space 183
5.4. Eigenvalue Analysis 192
5.5. Some Special Properties of Eigenvalues 197
5.6. Calculation of Eigenvalues 202
Problems 209
Further Reading 216
Chapter 6. Perfect Matrices 218
6.1. Synopsis 218
6.2. Implications of the Spectral Resolution Theorem 219
6.3. Diagonalization by a Similarity Transformation 226
6.4. Matrices with Distinct Eigenvalues 232
6.5. Unitary and Orthogonal Matrices 233
6.6. Semidiagonalization Theorem 238
6.7. Self-Adjoint Matrices 240
6.8. Normal Matrices 258
6.9. Miscellanea 262
6.10. The Initial Value Problem 267
6.11. Perturbation Theory 272
Problems 274
Further Reading 291
Chapter 7. Imperfect or Defective Matrices 292
7.1. Synopsis 292
7.2. Rank of the Characteristic Matrix 293
7.3. Jordan Block Diagonal Matrices 295
7.4. The Jordan Canonical Form 301
7.5. Determination of Generalized Eigenvectors 307
7.6. Dyadic Form of an Imperfect Matrix 316
7.7. Schmidt's Normal Form of an Arbitrary Square Matrix 317
7.8. The Initial Value Problem 321
Problems 323
Further Reading 327
Chapter 8. Infinite-Dimensional Linear Vector Spaces 328
8.1. Synopsis 328
8.2. Infinite-Dimensional Spaces 329
8.3. Riemann and Lebesgue Integration 332
8.4. Inner Product Spaces 335
8.5. Hilbert Spaces 337
8.6. Basis Vectors 339
8.7. Linear Operators 343
8.8. Solutions to Problems Involving ke-term Dyadics 349
8.9. Perfect Operators 356
Problems 364
Further Reading 366
Chapter 9. Linear Integral Operators in a Hilbert Space 368
9.1. Synopsis 368
9.2. Solvability Theorems 369
9.3. Completely Continuous and Hilbert-Schmidt Operators 379
9.4. Volterra Equations 388
9.5. Spectral Theory of Integral Operators 400
Problems 419
Further Reading 424
Chapter 10. Linear Differential Operators in a Hilbert Space 426
10.1. Synopsis 426
10.2. The Differential Operator 429
10.3. The Adjoint of a Differential Operator 433
10.4. Solution to the General Inhomogeneous Problem 439
10.5. Green's Function: Inverse of a Differential Operator 452
10.6. Spectral Theory of Differential Operators 465
10.7. Spectral Theory of Regular Sturm-Liouville Operators 472
10.8. Spectral Theory of Singular Sturm-Liouville Operators 490
10.9. Partial Differential Equations 507
Problems 515
Further Reading 523
APPENDIX 524
A.1. Section 3.2: Gauss Elimination and the Solution to the Linear System Ax = b 524
A.2. Example 3.6.1: Mass Separation with a Staged Absorber 527
A.3. Section 3.7: Iterative Methods for Solving the Linear System Ax = b 528
A.4. Exercise 3.7.2: Iterative Solution to Ax = b„Conjugate Gradient Method 531
A.5. Example 3.8.1: Convergence of the Picard and Newton-Raphson Methods 532
A.6. Example 3.8.2: Steady-State Solutions for a Continuously Stirred Tank Reactor 534
A.7. Example 3.8.3: The Density Profile in a Liquid-Vapor Interface (Iterative Solution of an Integral Equation) 536
A.8. Example 3.8.4: Phase Diagram of a Polymer Solution 539
A.9. Section 4.3: Gauss-Jordan Elimination and the Solution to the Linear System Ax = b 542
A.10. Section 5.4: Characteristic Polynomials and the Traces of a Square Matrix 544
A.11. Section 5.6: Iterative Method for Calculating the Eigenvalues of Tridiagonal Matrices 546
A.12. Example 5.6.1: Power Method for Iterative Calculation of Eigenvalues 547
A.13. Example 6.2.1: Implementation of the Spectral Resolution Theorem„Matrix Functions 548
A.14. Example 9.4.2: Numerical Solution of a Volterra Equation (Saturation in Porous Media) 550
A.15. Example 10.5.3: Numerical Green's Function Solution to a Second-Order Inhomogeneous Equation 553
A.16. Example 10.8.2: Series Solution to the Spherical Diffusion Equation (Carbon in a Cannonball) 555
Index 556

Produkt-Details

• Erscheint lt. Verlag: 12.7.2000
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Chemie
• Themenwelt: Technik ► Bauwesen
• ISBN-10: 0-08-051024-8 / 0080510248
• ISBN-13: 978-0-08-051024-8 / 9780080510248