Multidimensional Analysis
Algebras and Systems for Science and Engineering
Published on 22. August 2012
Book
Paperback/Softback
XII, 236 pages
978-1-4612-8697-4 (ISBN)
Description
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.
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.
Reviews / Votes
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."
More details
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Research
Illustrations
XII, 236 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 14 mm
Weight
388 gr
ISBN-13
978-1-4612-8697-4 (9781461286974)
DOI
10.1007/978-1-4612-4208-6
Schweitzer Classification
Other editions
Additional editions
Book
03/1995
Springer
€53.49
Shipment within 5-7 days
Content
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.- 1.2 The Mathematics of Dimensioned Quantities.- 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.- 4.2 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.- 6.2 Representing and Manipulating Dimensioned Scalars.- 6.3 Dimensioned Vectors.- 6.4 Representing Dimensioned Matrices.- 6.5 Operations on DimensionedMatrices.- 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.