A linear-algebraic formulation of dimensional analysis with constraints

Abstract

Dimensional analysis, especially Buckingham's $π$ theorem, reduces the number of variables by rewriting a relation in terms of dimensionless quantities. When variables are tied by definitions, constitutive laws, or other constraints, however, eliminating variables in advance can be awkward. We formulate dimensional analysis with constraints as linear algebra in logarithmic variables. Dimensional transformations and constraints are represented by subspaces, the effective number of independent dimensionless quantities is characterized by their intersection, and a matrix representation yields a systematic redundancy elimination procedure. Examples from falling motion, drag force, and stock-market indicators illustrate the scope and limitations of the method.