George W. Hart

Multidimensional Analysis (eBook, PDF)

Algebras and Systems for Science and Engineering

• Format: PDF

Produktbeschreibung

This book deals with the mathematical properties of dimensioned quantities, such as length, mass, voltage, and viscosity. Beginning with a careful examination of how one expresses the numerical results of a measurement and uses these results in subsequent manipulations, the author rigorously constructs the notion of dimensioned numbers and discusses their algebraic structure. The result is a unification of linear algebra and traditional dimensional analysis that can be extended from the scalars to which the traditional analysis is perforce restricted to multidimensional vectors of the sort frequently encountered in engineering, systems theory, economics, and other applications.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 236
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461242086
• Artikelnr.: 43992047

Inhaltsangabe

0. Introductory.- 0.1 Physical Dimensions.- 0.2 Mathematical Dimensions.- 0.3 Overview.- Exercises.- 1. The Mathematical Foundations of Science and Engineering.- 1.1 The Inadequacy of Real Numbers.- 1.1.1 The Error of Substitution.- 1.1.2 The Problem with Linear Spaces.- 1.1.3 Nondimensionalization.- 1.1.3 Dimensioned Algebras.- 1.2 The Mathematics of Dimensioned Quantities.- 1.2.1 Axiomatic Development.- 1.2.2 Constructive Approach.- 1.2.3 Constraints on Exponentiation.- 1.2.4 The Dimensional Basis.- 1.2.5 Dimensional Logarithms.- 1.2.6 The Basis-Independence Principle.- 1.2.7 Symmetries of Dimensioned Quantities.- 1.2.8 Images.- 1.3 Conclusions.- Exercises.- 2. Dimensioned Linear Algebra.- 2.1 Vector Spaces and Linear Transformations.- 2.2 Terminology and Dimensional Inversion.- 2.3 Dimensioned Scalars.- 2.4 Dimensioned Vectors.- 2.5 Dimensioned Matrices.- Exercises.- 3. The Theory of Dimensioned Matrices.- 3.1 The Dimensional Freedom of Multipliable Matrices.- 3.2 Endomorphic Matrices and the Matrix Exponential.- 3.3 Square Matrices, Inverses, and the Determinant.- 3.4 Squarable Matrices and Eigenstructure.- 3.5 Dimensionally Symmetric Multipliable Matrices.- 3.6 Dimensionally Hankel and Toeplitz Matrices.- 3.7 Uniform, Half Uniform, and Dimensionless Matrices.- 3.8 Conclusions.- 3.A Appendix: The n + m ? 1 Theorem.- Exercises.- 4. Norms, Adjoints, and Singular Value Decomposition.- 4.1 Norms for Dimensioned Spaces.- 4.1.1 Wand Norms.- 4.1.2 Extrinsic Norms.- 4.2 Dimensioned Singular Value Decomposition (DSVD).- 4.3 Adjoints.- 4.4 Norms for Nonuniform Matrices.- 4.5 A Control Application.- 4.6 Factorization of Symmetric Matrices.- Exercises.- 5. Aspects of the Theory of Systems.- 5.1 Differential and Difference Equations.- 5.2 State-Space Forms.- 5.3 Canonical Forms.- 5.4 Transfer Functions and Impulse Responses.- 5.5 Duals and Adjoints.- 5.6 Stability.- 5.7 Controllability, Observability, and Grammians.- 5.8 Expectations and Probability Densities.- Exercises.- 6. Multidimensional Computational Methods.- 6.1 Computers and Engineering.- 6.1.1 A Software Environment for Dimensioned Linear Algebra.- 6.1.2 Overview.- 6.2 Representing and Manipulating Dimensioned Scalars.- 6.2.1 The Numeric and Dimensional Components of a Scalar.- 6.2.2 The Dimensional Basis.- 6.2.3 Numerical Representations and Uniqueness.- 6.2.4 Scalar Operations.- 6.2.5 Input String Conversion.- 6.2.6 Output and Units Conversion.- 6.2.7 Binary Relations.- 6.2.8 Summary of Scalar Methods.- 6.3 Dimensioned Vectors.- 6.3.1 Dimensioned Vectors and Dimension Vectors.- 6.3.2 Representing Dimensioned Vectors.- 6.3.3 Vector Operations.- 6.3.4 Summary of Vectors.- 6.4 Representing Dimensioned Matrices.- 6.4.1 Arrays versus Matrices.- 6.4.2 The Domain/Range Matrix Representation.- 6.1.3 Allowing Geometric and Matrix Algebra Interpretations.- 6.4.4 Input Conversion.- 6.4.5 Output Conversion.- 6.4.6 Special Classes of Dimensioned Matrices.- 6.4.7 Identity and Zero Matrices.- 6.4.8 Scalar and Vector Conversion to Matrices.- 6.4.9 Summary of the Matrix Representation.- 6.5 Operations on Dimensioned Matrices.- 6.5.1 Matrix Addition, Subtraction, Similarity, and Equality.- 6.5.2 Block Matrices.- 6.5.3 Matrix Multiplication.- 6.5.4 Gaussian Elimination.- 6 5.5 The Determinant and Singularity.- 6.5.6 The Trace.- 6.5.7 Matrix Inverse.- 6.5.8 Matrix Transpose.- 6.5.9 Eigenstructure Decomposition.- 6.5.10 Singular Value Decomposition.- 6.6 Conclusions.- Exercises.- 7. Forms of Multidimensional Relationships.- 7.1 Goals.- 7.2 Operations.- 7.3 Procedure.- Exercises.- 8. Concluding Remarks.- 9. Solutions to Odd-Numbered Exercises.- References.

Rezensionen

FROM THE REVIEWS:

ZENTRALBLATT MATH
"As the book is addressed to physicists, engineers, economists and scientists in a broad sense, who are using linear algebra and linear systems theory in the multidimensional setting, the potential reader will find here a consistent and clear exposition of basic ideas and applications which meet their needs. To the more mathematically minded readers the book offers interesting structures and concepts that deserve generalization."