A Matrix Generalisation of Dimensional Analysis: New Similarity Transforms to Address the Problem of Uniqueness
Michael J. H. Taylor, Ángeles I. Díaz, Lucas A. Jodar Sanchez, Rafael Villanueva Micó · UEA Digital Repository (University of East Anglia) · 2008
On the verge of the centenary of dimensional analysis (DA), we present a new matrix generalisation of the Buckingham Theorem on which it is based. The proof is based on a solution we have found for inverting non-square block matrices and gives rise naturally to a new pair of transforms- the similarity transform (S) that converts physical dimensional data into dimensionless space and its inverse (S’). Although it is well known that DA: a) reduces the number of free parameters, 980 Taylor et al. b) guarantees scale invariance through dimensional homogeneity and c) extracts functional information encoded in the dimensionless grouping of variables, scientists seem to be unaware that the scaling laws provided by DA are degenerate and therefore not unique. We demonstrate that the inverse transform S ’ is responsible for the non-uniqueness and we show how reference to observational data is sufficient to to break the degeneracy inherent in transforming back to dimensional (physical)