Dimensional analysis is the study of relations among physical quantities through their dimensions. A physically meaningful equation must be dimensionally homogeneous, meaning that terms that are added or equated have the same dimensions.
If a relation contains quantities 
whose exponents in 
base dimensions are the columns of a 
matrix 
, then a product 
is dimensionless exactly when
 |
Thus finding dimensionless products is a null space problem. The Buckingham Pi theorem states
that if 
has matrix rank 
, a dimensionally homogeneous relation can be reduced to one
among 
independent dimensionless products. This algebraic structure is why dimensional analysis
is part of applied mathematics as well as a tool in physics and engineering.
