Matrices in Engineering Problems

Description

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