Multidimensional Analysis

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.

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.

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.

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.