Complete Set of Dimensionless Products

## Dimensionless products and complete set of dimensionless products

Recall that in dimensional analysis the product(ibid. 3.) is given by the expression

y = x1k1x2k1 ⋅ … ⋅ xnkn
If the product has no dimension it is called dimensionless product. In dimensional analysis the symbol π is used as a convention to denote dimensionless products. It does not equate to the number ≈ 3.14159.

To be more precise, π is used to symbolize the complete set of dimensionless products πi of a function.

π1 = x1k1'x2k1' ⋅ … ⋅ xnkn'
π2 = x1k1''x2k1'' ⋅ … ⋅ xnkn''

πp = x1k1px2k1p ⋅ … ⋅ xnknp
π = {π1, π2, …, πp}
Recall that,
The (product) function
y = ƒ (x1, x2, …, xn)
= x1k1x2k1 ⋅ … ⋅ xnkn
is dimensionally homogeneous if, and only if, the exponents k1, k2, …, kn are a solution of the linear equations
a = a1k1 + a2k2 + … + ankn
b = b1k1 + b2k2 + … + bnkn

g = g1k1 + g2k2 + … + gnkn
where a, b, …, g are the the dimensional exponents.
Therefore,
The product πi = x1k1ix2k1i ⋅ … ⋅ xnkni is dimensionless if, and only if, the exponents kji of the independent variables xj are the solution of
0 = a1k1i + a2k2i + … + ankni
0 = b1k1i + b2k2i + … + bnkni

0 = g1k1i + g2k2i + … + gnkni
where j = 1 to n the total number of independent variables, and the coefficients aj, bj, …, gj are the dimensional exponents making up rows in the dimensional matrix of the x's.
The set of dimensionless products is said to be complete if
A set of dimensionless products of given variables is complete, if each product in the set is independent of the others, and evey other dimensionless product of the variables is a product of powers of dimensionless products in the set.
The exponents of x's can be displayed in a matrix
x1x2 xn
π1k1'k2' kn'
π2k1''k2'' kn''
πpk1pk2p knp
This is referred to as the dimensional matrix of exponents.

## Independent dimensionless products

The products π1, π2, …, πp are said to be independent, if other than h1 = h2 = … = hp = 0 there are no constants h1, h2, …, hp such that
π1h1π2h2 ⋅ … ⋅ πphp ≡ 1
This is a special case of the definition of linear dependence. This may be defined from the perspective of linear algebra. But before that we define the linear combination of rows in a matrix as
If there exist constants corresponding to several rows of a matrix such that the sum of the products of several rows with their respective constants is another row of the matrix, that row is said to be a linear combination of other rows.
Then,
The rows of a matrix are said to be linearly dependent if there exists at least one row that is a linear combination of other rows.

In the matrix of exponents, this means that there exists constants h1, h2, …, hp where not all are zero such that, for a given column the sum of the products of a row element by its corresponding constant is zero, and this is so for all columns. Symbolically this means,
h1ki' + h2ki'' + … + hpkip = 0, for i = 1, 2, …, n.
Notice that for a matrix of two rows linear dependency is equivalent to a proportion between two rows. Therefore,
Linear dependecy is a generalization of the concept of proportionality.
From the defintion of linear dependency it follows that
A necessary and sufficient condition that the products π1, π2, …, πp be independent is that the rows in the matrix of exponents be linearly independent.
Hypothesis: Dimensionless products are independent. That is,
π1h1π2h2 ⋅ … ⋅ πphp ≡ 1 iff, hi = 0, for i = 1, 2, …, p.
And, the rows of matrix of exponents are linearly dependent.

We know that the matrix of exponents is composed from

π1 = x1k1'x2k1' ⋅ … ⋅ xnkn'
π2 = x1k1''x2k1'' ⋅ … ⋅ xnkn''

πp = x1k1px2k1p ⋅ … ⋅ xnknp .
Multiplying them we get,
π1π2 ⋅ … ⋅ πp = x1(k1' + k1'' + … + k1p)x2(k2' + k2'' + … + k2p) ⋅ … ⋅ xn(kn' + kn'' + … + knp).

Definition of linear dependency tell us that there must exist constants h1, h2, …, hp where not all are zero such that,

h1ki' + h2ki'' + … + hpkip = 0, for i = 1, 2, …, n.
Taking the power of each dimensionless product π1, π2, …, πp by its corresponding constant h1, h2, …, hp the above equation becomes,
π1h1π2h2 ⋅ … ⋅ πphp = x1(h1k1' + h1k1'' + … + h1k1p)x2(h2k2' + h2k2'' + … + h2k2p) ⋅ … ⋅ xn(hpkn' + hpkn'' + … + hpknp)
which gets reduced to
π1h1π2h2 ⋅ … ⋅ πphp = x10x20 ⋅ … ⋅ xn0.
Suggesting that
π1h1π2h2 ⋅ … ⋅ πphp = 1.
But, this is contrary to the hypothesis.

Therefore, the matrix of exponents must be linearly independent for the dimensionless products to be linearly independent.∎

Hypothesis: Rows in matrix of exponents are linearly independent. That is,
h1ki' + h2ki'' + … + hpkip = 0 iff, hi = 0, for i = 1, 2, …, n.
And, the dimensionless products are linearly dependent. That is, there exist constants h1, h2, …, hp where not all are zero such that,
π1h1π2h2 ⋅ … ⋅ πphp ≡ 1.

Substituting the expressions for π's shown belown

π1 = x1k1'x2k1' ⋅ … ⋅ xnkn'
π2 = x1k1''x2k1'' ⋅ … ⋅ xnkn''

πp = x1k1px2k1p ⋅ … ⋅ xnknp .
we get,
π1h1π2h2 ⋅ … ⋅ πphp = x1(h1k1' + h1k1'' + … + h1k1p)x2(h2k2' + h2k2'' + … + h2k2p) ⋅ … ⋅ xn(hpkn' + hpkn'' + … + hpknp).
But, from π1h1π2h2 ⋅ … ⋅ πphp ≡ 1 we have,
x1(h1k1' + h1k1'' + … + h1k1p)x2(h2k2' + h2k2'' + … + h2k2p) ⋅ … ⋅ xn(hpkn' + hpkn'' + … + hpknp) = 1
Suggesting that
i = 1 to p, j = 1 to n (hikj' + hikj'' + … + hikip) = 0.
and that,
h1ki' + h2ki'' + … + hpkip = 0, ∀i.
But, we assumed that not all constants h1, h2, …, hp are zero. Therefore, the observed result is contrary to the hypothesis.

Therefore, the condition that the matrix of exponents must be linearly independent is sufficient for the dimensionless products to be linearly independent.∎

## Computation of dimensionless products

Any fundamental system of solutions of
a1k1 + a2k2 + … + ankn = 0
b1k1 + b2k2 + … + bnkn = 0

g1k1 + g2k2 + … + gnkn = 0
furnishes exponents k1, k2, …, kn of a complete set of dimensionless products of the independent variables x1, x2, …, xn.

Conversely, the exponents k1, k2, …, kn of a complete set of dimensionless products of the independent variables x1, x2, …, xn are a fundamental system of solutions of
a1k1 + a2k2 + … + ankn = 0
b1k1 + b2k2 + … + bnkn = 0

g1k1 + g2k2 + … + gnkn = 0
The number of products in a complete set of dimensionless products of the independent variables x1, x2, …, xn is nr, in which r is the rank of the dimensional matrix of the variables.

We know from definition of a dimensionless product that the product has exponents k1, k2, …, kn that are a solution of

a1k1 + a2k2 + … + ankn = 0
b1k1 + b2k2 + … + bnkn = 0

g1k1 + g2k2 + … + gnkn = 0
Any such fundamental system of solutions will furnish linearly independent sets of exponents
k1', ki'', …, ki(n − r), for i = 1, 2, …, n
where r is the rank of the dimensional matrix of the coefficients .
Theorem
A necessary and sufficient condition that the products π1, π2, …, πp be independent it that the rows in the matrix of exponents be linearly independent.
tells us that because the exponents k1', ki'', …, ki(n − r) are linearly independent, they should yield independent dimensionless products.

Finally, because the above system of solutions possess no more than (nr) linearly independent solutions, it is impossible to have more than (nr) independent dimensionless products.