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## Über dieses Buch

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.

## Inhaltsverzeichnis

### 0. Introductory

Abstract
The Philosophy of Engineering, if such a field existed, would concern itself with broad questions about how models relate to reality and how our mathematical and computational tools manage to be so useful. One of the topics that discipline would surely investigate is the nature and representation of physical dimensions, such as “length,” “voltage,” and “viscosity.” If a consensus were reached on that topic, this book would be much shorter, as there would be a firm spot from which to begin a discussion of multidimensionality. However, there may be as many different conceptions of dimension as there are scientists and engineers. So, lacking a suitable starting point, this work deals with two topics:
i)
How should we model physically dimensioned quantities and their relationships?

ii)
How do linear algebra and multidimensional system models behave in the context of dimensioned quantities?

George W. Hart

### 1. The Mathematical Foundations of Science and Engineering

Abstract
The mathematics of scalar quantities in science and engineering has traditionally relied on the real and complex number systems. One theme of this chapter is that, in themselves, those number systems are not powerful enough to represent the algebraic structure that we need when we operate with physically dimensioned quantities. The main points are very simple, and the central proposals are not new in any essential way. I am simply trying to make explicit what many practitioners are already doing. The goal is to elucidate the unstated formal system that lies behind the use of dimensioned scalars. These initial arguments are necessary in order to have an agreed-upon foundation on which to build to more advanced issues of vectors and matrices in the following chapters.
George W. Hart

### 2. Dimensioned Linear Algebra

Abstract
Scientists and engineers use vectors and matrices in many aspects of their work and rely on a well-developed body of concepts and results from linear algebra. In this chapter, a careful examination of these applications shows that we actually use vectors and matrices that are best represented as n-tuples in which some or all of the elements carry physical dimensions (or “units”), such as volts, amperes, or meters. Many people have independently developed mathematical models of dimensioned scalar quantities along various lines similar to those in the previous chapter. However, to my knowledge, there are no previous examinations of the dimensioned vector and matrix structures considered here. Scientists, engineers, mathematicians, and philosophers have not questioned the use of traditional linear algebra in such circumstances. In fact, however, linear algebra was formally developed for dimensionless vectors and matrices, defined in terms of fields that are closed under addition, and in many respects it is not valid for multidimensional applications.
George W. Hart

### 3. The Theory of Dimensioned Matrices

Abstract
We have now seen the basic machinery of dimensioned linear algebra and how all multipliable matrices have the dimensional form A ~ ab~ in which the rows are dimensionally parallel and the columns are dimensionally parallel. This chapter examines many special subclasses within this set of multipliable matrices and demonstrates how their form relates to their function.
George W. Hart

### 4. Norms, Adjoints, and Singular Value Decomposition

Abstract
Norms for measuring vectors and matrices, adjoints, and the singular value decomposition (SVD) are all areas where traditional methods are basis-dependent and/or dimensionally inhomogeneous. The methods presented in this chapter correct these problems.
George W. Hart

### 5. Aspects of the Theory of Systems

Abstract
System theory is a premier application of linear algebra. It is concerned with developing, applying, and understanding the properties of mathematical models of dynamic systems. Although “multidimensional systems” have been an important and rich object of study, system theory has historically maintained a mathematical perspective that ignores the complexities of true physical dimensionality. The expression multidimensional system has been used to signify the quantity, not quality, of the signal components.
George W. Hart

### 6. Multidimensional Computational Methods

Abstract
Scientists and engineers wishing to compute with the mathematical objects and operations of dimensioned linear algebra require software tools somewhat different from the tools currently available. The study of computational methods for dimensioned quantities can be called multidimensional methods, analogous to traditional numerical methods. This chapter presents compact data structures for representing dimensioned quantities and efficient algorithms for performing the standard operations of linear algebra. Manipulations such as products, inverses, eigenstructure decomposition, singular value decomposition can be carried out with dimensioned scalars, vectors, and matrices, but we need efficient algorithms to check constraints and compute dimensions. Although the dimensional structures may be more intricate, it is shown that the space and time complexities required for dimensioned operations are no higher than the complexities of the corresponding standard algorithms for dimensionless quantities. Thus, there is no significant computational burden added due to the dimensioned nature of the matrices, for large matrices at least.
George W. Hart

### 7. Forms of Multidimensional Relationships

Abstract
This chapter briefly considers the question of extending the traditional scalar methods of dimensional analysis,56 used for deriving the forms of scalar relationships, to methods for deriving the forms of multidimensional equations; it is intended to be exploratory and provocative. As a multidimensional form of dimensional analysis, we want to develop techniques of multidimensional analysis in which the dimensional structures of vectors and matrices are used not just in checking consistency and expressing dimensional forms, but also for deriving new results.
George W. Hart

### 8. Concluding Remarks

Abstract
We have begun to explore a new branch of applied mathematics, which includes multidimensional analysis and dimensioned linear algebra, and we have discovered a rich structure of interrelated concepts. The most fundamental aspect of this work is that it harmonizes the previously unconnected ideas of mathematical dimensions and physical dimensions. Scientists and engineers need to work with both notions of multidimensionality simultaneously, and the previously available mathematical tools were not suitable for the task. It is not a linguistic accident that the single term dimension was applied to both of these concepts. The dimension vectors defined here are a form of generalized dimension that naturally subsumes the different kinds of degrees of freedom found in the two traditional notions of dimension.
George W. Hart

### 9. Solutions to Odd-Numbered Exercises

Abstract
With x a length: (a) the Taylor series $$1 + x + {x^2} + {x^3} + \ldots$$ is the sum of a dimensionless quantity, a length, an area, a volume, etc.; (b) the formula 1 + x/n sums a dimenslonless quantity and a length; (c) the derivative $$\frac{d}{{dx}}\,f(x)$$ has dimensions of [f/length] and so can not equal [f]; (d) the condition that [f]=[f2] requires that f be dimensionless, but according to a result in §1.2.6, there can be no intrinsic function from lengths to dimensionless quantities.
George W. Hart

### Backmatter

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