Given a system of n dimensional variables q 1 , … … , q n in ℓ ℓ fundamental (basis) dimensions, the dimensional matrix is the ℓ ℓ × × n matrix M whose ℓ ℓ rows correspond to the fundamental dimensions and whose n columns are the dimensions of the variables: the ( i , j ) th entry (where 1 ≤ ≤ i ≤ ≤ ℓ ℓ and 1 ≤ ≤ j ≤ ≤ n ) is the power of the i th fundamental dimension in the j th variable. The matrix can be interpreted as taking in a combination of the variable quantities and giving out the dimensions of the combination in terms of the fundamental dimensions. So the ℓ ℓ × × 1 (column) vector that results from the multiplication M [ a 1 ⋮ ⋮ a n ] consists of the units of q 1 a 1 q 2 a 2 ⋯ ⋯ q n a n in terms of the ℓ ℓ fundamental independent (basis) units.