This paper presents a concrete, algebraic way to do dimensional analysis when the variables are linked by extra equations or definitions. The author turns the usual multiplicative relations between physical quantities into linear relations by taking logarithms. That change makes it possible to count independent dimensionless combinations and to remove redundant ones by routine matrix operations instead of guesswork.

The basic idea is familiar: a physical quantity’s dimensional powers form a matrix A. Combinations that stay the same when units change are exponent vectors in the kernel of A. By passing to logarithmic variables, monomials become inner products and scaling transformations become simple translations. In that language the classical Buckingham π result — the number of independent dimensionless groups equals n − rank(A) — appears as a statement about the dimension of ker(A).

The new work adds constraints among the variables. These constraints are written as equations ψ(y)=0 in the logarithmic variables, and their linear approximation is the Jacobian matrix J. The allowed dimensionless changes are then the intersection of ker(A) and ker(J). In other words, the effective number of admissible independent dimensionless groups equals the dimension of ker(A) ∩ ker(J). For constraints that are scale invariant (the common case when a constraint does not change under a unit rescaling), this simplifies to deff = n − rank(A) − rank(J).

Beyond counting, the paper gives a step-by-step algebraic way to find independent groups and drop redundant ones. One chooses a basis E for ker(A) and forms the vector of candidate π-groups. The Jacobian J is used to build a smaller matrix C = J E (E^T E)^{-1}. Linear relations among the candidate π-groups appear as linear relations among rows of C, so row reduction of C identifies which candidates are redundant and yields a reduced independent set. The author lists these steps explicitly as a practical procedure.