Elements of Matrix Modeling and Computing with MATLAB

This book covers a number of topics in linear algebra that are often not covered in first-year calculus, such as complex numbers and functions, matrices, algebraic systems, curve fitting, elements of linear differential equations, transform methods, and tools of computation. It introduces 2- and 3-D vectors and their products, connecting them to multivariate calculus using geometric and algebraic approaches. The author illuminates concepts through several applications, including steady state models, mixing tanks, heat diffusion, and pin-jointed structures. The text also incorporates exercises and MATLAB to help with more complex calculations.

As discrete models and computing have become more common, there is a need to study matrix computation and numerical linear algebra. Encompassing a diverse mathematical core, Elements of Matrix Modeling and Computing with MATLAB examines a variety of applications and their modeling processes, showing you how to develop matrix models and solve algebraic systems. Emphasizing practical skills, it creates a bridge from problems with two and three variables to more realistic problems that have additional variables. Elements of Matrix Modeling and Computing with MATLAB focuses on seven basic applications: circuits, trusses, mixing tanks, heat conduction, data modeling, motion of a mass, and image filters. These applications are developed from very simple to more complex models. To explain the processes, the book explores numerous topics in linear algebra, including complex numbers and functions, matrices, algebraic systems, curve fitting, elements of linear differential equations, transform methods, and tools of computation. For example, the author uses linearly independent vectors and subspaces to explain over- and under-determined systems, eigenvalues and eigenvectors to solve initial value problems, and discrete Fourier transforms to perform image filtering in the frequency domain. Although the primary focus is to cultivate calculation skills by hand, most chapters also include MATLAB to help with more complicated calculations.

List of FiguresList of TablesPrefaceIntroductionVECTORS IN THE PLANE Floating Point and Complex NumbersComplex Valued Functions Vectors in R2Dot Product and Work Lines and Curves in R2 and C VECTORS IN SPACE Vectors and Dot Product Cross and Box Products Lines and Curves in R3 Planes in R3 Extensions to Rn Ax = d: UNIQUE SOLUTION Matrix ModelsMatrix Products Special Cases of Ax = d Row Operations and Gauss EliminationInverse Matrices LU Factorization Determinants and Cramer's RuleAx = d: LEAST SQUARES SOLUTIONCurve Fitting to DataNormal EquationsMultilinear Data Fitting Parameter IdentificationAx = d: MULTIPLE SOLUTIONSSubspaces and Solutions in R3Row Echelon FormNullspaces and Equilibrium Equations LINEAR INITIAL VALUE PROBLEMS First Order Linear Second Order Linear Homogeneous and Complex Solution Nonhomogeneous Linear Differential Equations System Form of Linear Second OrderEIGENVALUES AND DIFFERENTITAL EQUATIONS Solution of x' = Ax by Elimination Real Eigenvalues and Eigenvectors Solution of x' = Ax + f (t)IMAGE PROCESSING IN THE SPACE DOMAIN Matrices and Images Contrast and HistogramsBlurring and Sharpening IMAGE PROCESSING IN THE FREQUENCY DOMAIN Laplace and Fourier Transforms Properties of DFT DFT in Rn × Rn Frequency Filters in Rn × RnAppendix: Solutions to Odd ExercisesBibliographyIndex