Linear Algebra and Linear Operators in Engineering: With Applications in Mathematica®
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Table of contents
- Linear Algebra and Linear Operators in Engineeringiii
- Copyright Pageiv
- Contentsv
- Prefacexi
- Chapter 1. Determinants1
- 1.1. Synopsis1
- 1.2. Matrices2
- 1.3. Definition of a Determinant3
- 1.4. Elementary Properties of Determinants6
- 1.5. Cofactor Expansions9
- 1.6. Cramer's Rule for Linear Equations14
- 1.7. Minors and Rank of Matrices16
- Problems18
- Further Reading22
- Chapter 2. Vectors and Matrices25
- 2.1. Synopsis25
- 2.2. Addition and Multiplication26
- 2.3. The Inverse Matrix28
- 2.4. Transpose and Adjoint33
- 2.5. Partitioning Matrices35
- 2.6. Linear Vector Spaces38
- Problems43
- Further Reading46
- Chapter 3. Solution of Linear and Nonlinear Systems47
- 3.1. Synopsis47
- 3.2. Simple Gauss Elimination48
- 3.3. Gauss Elimination with Pivoting55
- 3.4. Computing the Inverse of a Matrix58
- 3.5. LU-Decomposition61
- 3.6. Band Matrices66
- 3.7. Iterative Methods for Solving Ax = b78
- 3.8. Nonhnear Equations85
- Problems108
- Further Reading121
- Chapter 4. General Theory of Solvability of Linear Algebraic Equations123
- 4.1. Synopsis123
- 4.2. Sylvester's Theorem and the Determinants of Matrix Products124
- 4.3. Gauss-Jordan Transformation of a Matrix129
- 4.4. General Solvability Theorem for Ax = b133
- 4.5. Linear Dependence of a Vector Set and the Rank of Its Matrix150
- 4.6. The Fredholm Alternative Theorem155
- Problems159
- Further Reading161
- Chapter 5. The Eigenproblem163
- 5.1. Synopsis163
- 5.2. Linear Operators in a Normed Linear Vector Space165
- 5.3. Basis Sets in a Normed Linear Vector Space170
- 5.4. Eigenvalue Analysis179
- 5.5. Some Special Properties of Eigenvalues184
- 5.6. Calculation of Eigenvalues189
- Problems196
- Further Reading203
- Chapter 6. Perfect Matrices205
- 6.1. Synopsis205
- 6.2. Implications of the Spectral Resolution Theorem206
- 6.3. Diagonalization by a Similarity Transformation213
- 6.4. Matrices with Distinct Eigenvalues219
- 6.5. Unitary and Orthogonal Matrices220
- 6.6. Semidiagonalization Theorem225
- 6.7. Self-Adjoint Matrices227
- 6.8. Normal Matrices245
- 6.9. Miscellanea249
- 6.10. The Initial Value Problem254
- 6.11. Perturbation Theory259
- Problems261
- Further Reading278
- Chapter 7. Imperfect or Defective Matrices279
- 7.1. Synopsis279
- 7.2. Rank of the Characteristic Matrix280
- 7.3. Jordan Block Diagonal Matrices282
- 7.4. The Jordan Canonical Form288
- 7.5. Determination of Generalized Eigenvectors294
- 7.6. Dyadic Form of an Imperfect Matrix303
- 7.7. Schmidt's Normal Form of an Arbitrary Square Matrix304
- 7.8. The Initial Value Problem308
- Problems310
- Further Reading314
- Chapter 8. Infinite-Dimensional Linear Vector Spaces315
- 8.1. Synopsis315
- 8.2. Infinite-Dimensional Spaces316
- 8.3. Riemann and Lebesgue Integration319
- 8.4. Inner Product Spaces322
- 8.5. Hilbert Spaces324
- 8.6. Basis Vectors326
- 8.7. Linear Operators330
- 8.8. Solutions to Problems Involving ke-term Dyadics336
- 8.9. Perfect Operators343
- Problems351
- Further Reading353
- Chapter 9. Linear Integral Operators in a Hilbert Space355
- 9.1. Synopsis355
- 9.2. Solvability Theorems356
- 9.3. Completely Continuous and Hilbert-Schmidt Operators366
- 9.4. Volterra Equations375
- 9.5. Spectral Theory of Integral Operators387
- Problems406
- Further Reading411
- Chapter 10. Linear Differential Operators in a Hilbert Space413
- 10.1. Synopsis413
- 10.2. The Differential Operator416
- 10.3. The Adjoint of a Differential Operator420
- 10.4. Solution to the General Inhomogeneous Problem426
- 10.5. Green's Function: Inverse of a Differential Operator439
- 10.6. Spectral Theory of Differential Operators452
- 10.7. Spectral Theory of Regular Sturm-Liouville Operators459
- 10.8. Spectral Theory of Singular Sturm-Liouville Operators477
- 10.9. Partial Differential Equations494
- Problems502
- Further Reading510
- APPENDIX511
- A.1. Section 3.2: Gauss Elimination and the Solution to the Linear System Ax = b511
- A.2. Example 3.6.1: Mass Separation with a Staged Absorber514
- A.3. Section 3.7: Iterative Methods for Solving the Linear System Ax = b515
- A.4. Exercise 3.7.2: Iterative Solution to Ax = b„Conjugate Gradient Method518
- A.5. Example 3.8.1: Convergence of the Picard and Newton-Raphson Methods519
- A.6. Example 3.8.2: Steady-State Solutions for a Continuously Stirred Tank Reactor521
- A.7. Example 3.8.3: The Density Profile in a Liquid-Vapor Interface (Iterative Solution of an Integr523
- A.8. Example 3.8.4: Phase Diagram of a Polymer Solution526
- A.9. Section 4.3: Gauss-Jordan Elimination and the Solution to the Linear System Ax = b529
- A.10. Section 5.4: Characteristic Polynomials and the Traces of a Square Matrix531
- A.11. Section 5.6: Iterative Method for Calculating the Eigenvalues of Tridiagonal Matrices533
- A.12. Example 5.6.1: Power Method for Iterative Calculation of Eigenvalues534
- A.13. Example 6.2.1: Implementation of the Spectral Resolution Theorem„Matrix Functions535
- A.14. Example 9.4.2: Numerical Solution of a Volterra Equation (Saturation in Porous Media)537
- A.15. Example 10.5.3: Numerical Green's Function Solution to a Second-Order Inhomogeneous Equation540
- A.16. Example 10.8.2: Series Solution to the Spherical Diffusion Equation (Carbon in a Cannonball)542
- Index543
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
· 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