Linear Algebra and Linear Operators in Engineering: With Applications in Mathematica® (Volume 3) (Process Systems Engineering, Volume 3)

✍ Scribed by H. Ted Davis, Kendall T. Thomson

• Publisher: Academic Press
• Year: 2000
• Tongue: English
• Leaves: 561
• Edition: 1
• Category: Library

✦ Synopsis

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

✦ Table of Contents

Linear Algebra and Linear Operators in Engineering
Copyright Page
Contents
Preface
Chapter 1. Determinants
1.1. Synopsis
1.2. Matrices
1.3. Definition of a Determinant
1.4. Elementary Properties of Determinants
1.5. Cofactor Expansions
1.6. Cramer's Rule for Linear Equations
1.7. Minors and Rank of Matrices
Problems
Further Reading
Chapter 2. Vectors and Matrices
2.1. Synopsis
2.2. Addition and Multiplication
2.3. The Inverse Matrix
2.4. Transpose and Adjoint
2.5. Partitioning Matrices
2.6. Linear Vector Spaces
Problems
Further Reading
Chapter 3. Solution of Linear and Nonlinear Systems
3.1. Synopsis
3.2. Simple Gauss Elimination
3.3. Gauss Elimination with Pivoting
3.4. Computing the Inverse of a Matrix
3.5. LU-Decomposition
3.6. Band Matrices
3.7. Iterative Methods for Solving Ax = b
3.8. Nonhnear Equations
Problems
Further Reading
Chapter 4. General Theory of Solvability of Linear Algebraic Equations
4.1. Synopsis
4.2. Sylvester's Theorem and the Determinants of Matrix Products
4.3. Gauss-Jordan Transformation of a Matrix
4.4. General Solvability Theorem for Ax = b
4.5. Linear Dependence of a Vector Set and the Rank of Its Matrix
4.6. The Fredholm Alternative Theorem
Problems
Further Reading
Chapter 5. The Eigenproblem
5.1. Synopsis
5.2. Linear Operators in a Normed Linear Vector Space
5.3. Basis Sets in a Normed Linear Vector Space
5.4. Eigenvalue Analysis
5.5. Some Special Properties of Eigenvalues
5.6. Calculation of Eigenvalues
Problems
Further Reading
Chapter 6. Perfect Matrices
6.1. Synopsis
6.2. Implications of the Spectral Resolution Theorem
6.3. Diagonalization by a Similarity Transformation
6.4. Matrices with Distinct Eigenvalues
6.5. Unitary and Orthogonal Matrices
6.6. Semidiagonalization Theorem
6.7. Self-Adjoint Matrices
6.8. Normal Matrices
6.9. Miscellanea
6.10. The Initial Value Problem
6.11. Perturbation Theory
Problems
Further Reading
Chapter 7. Imperfect or Defective Matrices
7.1. Synopsis
7.2. Rank of the Characteristic Matrix
7.3. Jordan Block Diagonal Matrices
7.4. The Jordan Canonical Form
7.5. Determination of Generalized Eigenvectors
7.6. Dyadic Form of an Imperfect Matrix
7.7. Schmidt's Normal Form of an Arbitrary Square Matrix
7.8. The Initial Value Problem
Problems
Further Reading
Chapter 8. Infinite-Dimensional Linear Vector Spaces
8.1. Synopsis
8.2. Infinite-Dimensional Spaces
8.3. Riemann and Lebesgue Integration
8.4. Inner Product Spaces
8.5. Hilbert Spaces
8.6. Basis Vectors
8.7. Linear Operators
8.8. Solutions to Problems Involving ke-term Dyadics
8.9. Perfect Operators
Problems
Further Reading
Chapter 9. Linear Integral Operators in a Hilbert Space
9.1. Synopsis
9.2. Solvability Theorems
9.3. Completely Continuous and Hilbert-Schmidt Operators
9.4. Volterra Equations
9.5. Spectral Theory of Integral Operators
Problems
Further Reading
Chapter 10. Linear Differential Operators in a Hilbert Space
10.1. Synopsis
10.2. The Differential Operator
10.3. The Adjoint of a Differential Operator
10.4. Solution to the General Inhomogeneous Problem
10.5. Green's Function: Inverse of a Differential Operator
10.6. Spectral Theory of Differential Operators
10.7. Spectral Theory of Regular Sturm-Liouville Operators
10.8. Spectral Theory of Singular Sturm-Liouville Operators
10.9. Partial Differential Equations
Problems
Further Reading
APPENDIX
A.1. Section 3.2: Gauss Elimination and the Solution to the Linear System Ax = b
A.2. Example 3.6.1: Mass Separation with a Staged Absorber
A.3. Section 3.7: Iterative Methods for Solving the Linear System Ax = b
A.4. Exercise 3.7.2: Iterative Solution to Ax = b—Conjugate Gradient Method
A.5. Example 3.8.1: Convergence of the Picard and Newton-Raphson Methods
A.6. Example 3.8.2: Steady-State Solutions for a Continuously Stirred Tank Reactor
A.7. Example 3.8.3: The Density Profile in a Liquid-Vapor Interface (Iterative Solution of an Integral Equation)
A.8. Example 3.8.4: Phase Diagram of a Polymer Solution
A.9. Section 4.3: Gauss-Jordan Elimination and the Solution to the Linear System Ax = b
A.10. Section 5.4: Characteristic Polynomials and the Traces of a Square Matrix
A.11. Section 5.6: Iterative Method for Calculating the Eigenvalues of Tridiagonal Matrices
A.12. Example 5.6.1: Power Method for Iterative Calculation of Eigenvalues
A.13. Example 6.2.1: Implementation of the Spectral Resolution Theorem—Matrix Functions
A.14. Example 9.4.2: Numerical Solution of a Volterra Equation (Saturation in Porous Media)
A.15. Example 10.5.3: Numerical Green's Function Solution to a Second-Order Inhomogeneous Equation
A.16. Example 10.8.2: Series Solution to the Spherical Diffusion Equation (Carbon in a Cannonball)
INDEX