  On the verge of the centenary of dimensional analysis (DA), we present a
generalisation of the theory and a methodology for the discovery of empirical
laws from observational data. It is well known that DA: a) reduces the number
of free parameters, b) guarantees scale invariance through dimensional
homogeneity and c) extracts functional information encoded in the dimensionless
grouping of variables. Less known are the results of Rudolph and co-workers
that DA also gives rise to a new pair of transforms - the similarity transform
(S) that converts physical dimensional data into dimensionless space and its
inverse (S'). Here, we present a new matrix generalisation of the Buckingham
Theorem, made possible by recent developments in the theory of inverse
non-square matrices, and show how the transform pair arises naturally. We
demonstrate that the inverse transform S' is non-unique and how this casts
doubt on scaling relations obtained in cases where observational data has not
been referred to in order to break the degeneracy inherent in transforming back
to dimensional (physical) space. As an example, we show how the underlying
functional form of the Planck Radiation Law can be deduced in only a few lines
using the matrix method and without appealing to first principles; thus
demonstrating the possibility of a priori knowledge discovery; but that
subsequent data analysis is still required in order to identify the exact
causal law. It is hoped that the proof presented here will give theoreticians
confidence to pursue inverse problems in physics using DA.
