Matrices in Engineering Problems

Book description


This book is intended as an undergraduate text introducing matrix methods as they relate to engineering problems. It begins with the fundamentals of mathematics of matrices and determinants. Matrix inversion is discussed, with an introduction of the well known reduction methods. Equation sets are viewed as vector transformations, and the conditions of their solvability are explored. Orthogonal matrices are introduced with examples showing application to many problems requiring three dimensional thinking. The angular velocity matrix is shown to emerge from the differentiation of the 3-D orthogonal matrix, leading to the discussion of particle and rigid body dynamics.

The book continues with the eigenvalue problem and its application to multi-variable vibrations. Because the eigenvalue problem requires some operations with polynomials, a separate discussion of these is given in an appendix. The example of the vibrating string is given with a comparison of the matrix analysis to the continuous solution.

Table of Contents: Matrix Fundamentals / Determinants / Matrix Inversion / Linear Simultaneous Equation Sets / Orthogonal Transforms / Matrix Eigenvalue Analysis / Matrix Analysis of Vibrating Systems

Table of contents

  1. Preface
  2. Matrix Fundamentals
    1. Definition of A Matrix
      1. Notation
    2. Elemetary Matrix Algebra
      1. Addition (Including Subtraction)
      2. Multiplication by A Scalar
      3. Vector Multiplication
      4. Matrix Multiplication
      5. Transposition
    3. Basic Types of Matrices
      1. The Unit Matrix
      2. The Diagonal Matrix
      3. Orthogonal Matrices
      4. Triangular Matrices
      5. Symmetric and Skew-Symmetric Matrices
      6. Complex Matrices
      7. The Inverse Matrix
    4. Transformation Matrices
    5. Matrix Partitioning
    6. Interesting Vector Products
      1. An Interpretation of Ax = c
      2. The (nX1X1Xn) Vector Product
      3. Vector Cross Product
    7. Examples
      1. An Example Matrix Multiplication
      2. An Example Matrix Triple Product
      3. Multiplication of Complex Matrices
    8. Exercises
  3. Determinants
    1. Introduction
    2. General Definition of a Determinant
    3. Permutations and Inversions of Indices
      1. Inversions
      2. An Example Determinant Expansion
    4. Properties of Determinants
    5. The Rank of a Determinant
    6. Minors and Cofactors
      1. Expansions by Minors—LaPlace Expansions
      2. Expansion by Lower Order Minors
      3. The Determinant of a Matrix Product
    7. Geometry: Lines, Areas, and Volumes
    8. The Adjoint and Inverse Matrices
      1. Rank of the Adjoint Matrix
    9. Determinant Evaluation
      1. Pivotal Condensation
      2. Gaussian Reduction
      3. Rank of the Determinant Less Than n
    10. Examples
      1. Cramer's Rule
      2. An Example Complex Determinant
      3. The ``Characteristic Determinant''
    11. Exercises
  4. Matrix Inversion
    1. Introduction
    2. Elementary Operations in Matrix Form
      1. Diagonalization Using Elementary Matrices
    3. Gauss-Jordan Reduction
      1. Singular Matrices
    4. The Gauss Reduction Method
      1. Gauss Reduction in Detail
      2. Example Gauss Reduction
    5. LU Decomposition
      1. LU Decomposition in Detail
      2. Example LU Decomposition
    6. Matrix Inversion By Partitioning
    7. Additional Topics
      1. Column Normalization
      2. Improving the Inverse
      3. Inverse of a Triangular Matrix
      4. Inversion by Orthogonalization
      5. Inversion of a Complex Matrix
    8. Examples
      1. Inversion Using Partitions
    9. Exercises
  5. Linear Simultaneous Equation Sets
    1. Introduction
    2. Vectors and Vector Sets
      1. Linear Independence of a Vector Set
      2. Rank of a Vector Set
    3. Simultaneous Equation Sets
      1. Square Equation Sets
      2. Underdetermined Equation Sets
      3. Overdetermined Equation Sets
    4. Linear Regression
      1. Example Regression Problem
      2. Quadratic Curve Fit
    5. Lagrange Interpolation Polynomials
      1. Interpolation
      2. The Lagrange Polynomials
    6. Exercises
  6. Orthogonal Transforms
    1. Introduction
    2. Orthogonal Matrices and Transforms
      1. Righthanded Coordinates, and Positive Angle
    3. Example Coordinate Transforms
      1. Earth-Centered Coordinates
      2. Rotation About a Vector (Not a Coordinate Axis)
      3. Rotation About all Three Coordinate Axes
      4. Solar Angles
      5. Image Rotation in Computer Graphics
    4. Congruent and Similarity Matrix Transforms
    5. Differentiation of Matrices, Angular Velocity
      1. Velocity of a Point on a Wheel
    6. Dynamics of a Particle
    7. Rigid Body Dynamics
      1. Rotation of a Rigid Body
      2. Moment of Momentum
      3. The Inertia Matrix
      4. The Torque Equation
    8. Examples
    9. Exercises
  7. Matrix Eigenvalue Analysis
    1. Introduction
    2. The Eigenvalue Problem
      1. The Characteristic Equation and Eigenvalues
      2. Synthesis of A by its Eigenvalues and Eigenvectors
      3. Example Analysis of a Nonsymmetric 3X3
      4. Eigenvalue Analysis of Symmetric Matrices
    3. Geometry of the Eigenvalue Problem
      1. Non-Symmetric Matrices
      2. Matrix with a Double Root
    4. The Eigenvectors and Orthogonality
      1. Inverse of the Characteristic Matrix
      2. Vibrating String Problem
    5. The Cayley-Hamilton Theorem
      1. Functions of a Square Matrix
      2. Sylvester's Theorem
    6. Mechanics of the Eigenvalue Problem
      1. Calculating the Characteristic Equation Coefficients
      2. Factoring the Characteristic Equation
      3. Calculation of the Eigenvectors
    7. Example Eigenvalue Analysis
      1. Example Eigenvalue Analysis; Complex Case
      2. Eigenvalues by Matrix Iteration
    8. The Eigenvalue Analysis of Similar Matrices; Danilevsky's Method
      1. Danilevsky's Method
      2. Example of Danilevsky's Method
      3. Danilevsky's Method—Zero Pivot
    9. Exercises
  8. Matrix Analysis of Vibrating Systems
    1. Introduction
    2. Setting up Equations, Lagrange's Equations
      1. Generalized Form of Lagrange's Equations
      2. Mechanical / Electrical Analogies
      3. Examples using the Lagrange Equations
    3. Vibration of Conservative Systems
      1. Conservative Systems – The Initial Value Problem
      2. Interpretation of Equation (7.23)
      3. Conservative Systems - Sinusoidal Response
      4. Vibrations in a Continuous Medium
    4. Nonconservative Systems. Viscous Damping
      1. The Initial Value Problem
      2. Sinusoidal Response
      3. Determining the Vector Coefficients for the Driven System
      4. Sinusoidal Response – NonZero Initial Conditions
    5. Steady State Sinusoidal Response
      1. Analysis of Ladder Networks; The Cumulant
    6. Runge-Kutta Integration of Differential Equations
    7. Exercises
  9. Partial Differentiation of Bilinear and Quadratic Forms
  10. Polynomials
    1. Polynomial Basics
    2. Polynomial Arithmetic
      1. Evaluating a Polynomial at a Aiven Value
    3. Evaluating Polynomial Roots
      1. The Laguerre Method
      2. The Newton Method
      3. An Example
  11. The Vibrating String
    1. The Digitized – Matrix Solution
    2. The Continuous Function Solution
    3. Exercises
  12. Solar Energy Geometry
    1. Yearly Energy Output
    2. An Example
    3. Tracking the Sun
  13. Answers to Selected Exercises
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  14. Bibliography
  15. Author's Biography
  16. Index

Product information

  • Title: Matrices in Engineering Problems
  • Author(s): Marvin Tobias
  • Release date: July 2011
  • Publisher(s): Morgan & Claypool Publishers
  • ISBN: 9781608456598