Transforming a set of equations from one set of variables to another? Assume $\theta_1,\dots,\theta_n$ are $n$ positive numbers such that $$\theta_1+\dots+\theta_n=1$$ Define $y_{ij}=\frac{\theta_i}{\theta_j}$ for all $i,j \in \{1,\dots,n\}$. Is there a way to transform the above equation in terms of $y_{ij}$. 
 A: No. If you multiply all $\theta_i$ by the same positive number $c\neq 1$, then all $y_{ij}$ stay the same, but $\theta_1+\dots+\theta_n$ changes.
In more details: assume that you have a condition or a set of conditions on $y_{ij}=\frac{\theta_i}{\theta_j}$ which is satisfied if and only if $\theta_1+\dots+\theta_n=1$. Let $\overline{\theta}_i$ be positive numbers such that $\overline{\theta}_1+\dots+\overline{\theta}_n=1$. Then, by assumption, the numbers $\overline{y}_{ij}=\frac{\overline{\theta}_i}{\overline{\theta}_j}$ satisfy that condition(s). Now choose a positive number $c\neq 1$ and set $\theta'_i = c\overline{\theta}_i$. Then $y'_{ij} = \frac{\theta'_i}{\theta'_j} = \overline{y}_{ij}$, so the numbers $y'_{ij}$ satisfy the condition(s) as well, but $\theta'_1+\dots+\theta'_n = c(\overline{\theta}_1+\dots+\overline{\theta}_n) = c\neq 1$.
A: Yes. Use $\theta_i = \theta_j y_{ij}$ to write $\theta_1+\theta_2\dots+\theta_n=1$ as $\theta_j y_{1j}+\theta_j y_{2j}+\dots+ \theta_j y_{nj}= 1$
Collect $\theta_j$ s.t. $\theta_j(y_{1j}+y_{2j}+\dots+y_{nj})= 1$ so $\theta_j = \displaystyle \frac{1}{y_{1j}+y_{2j}+\dots+y_{nj}}=\frac{1}{\displaystyle \sum_{k=1}^{n}y_{kj}}$
$\theta_1+\theta_2\dots+\theta_n=1 = \displaystyle \frac{1}{\displaystyle \sum_{k=1}^{n}y_{k1}}  + \frac{1}{\displaystyle \sum_{k=1}^{n}y_{k2}} + \dots+\frac{1}{\displaystyle \sum_{k=1}^{n}y_{kn}} = \displaystyle \sum_{j=1}^{n}\frac{1}{\displaystyle \sum_{k=1}^{n}y_{kj}}$

Edit 2
More simply:
Given
$$\theta_1 + \theta_2 + \dots+ \theta_‌​n= 1$$
$$\theta_j > 0$$
$$y_{ij} = \frac{\theta_i}{\theta_j}$$
then
$$ \theta_j = \displaystyle \frac{\theta_j}{\theta_1 + \theta_2 + \dots+ \theta_‌​n} = \displaystyle \frac{1}{\frac{\theta_1}{\theta_j}+\frac{\theta_2}{\theta_j}+\dots+\frac{\theta_‌​‌​n}{\theta_j}} = \displaystyle \frac{1}{y_{1j}+y_{2j}+\dots+y_{nj}}$$
Or
$$ y_{1j}+y_{2j}+\dots+y_{nj} = \displaystyle \frac{1}{\theta_j}$$
So each $\theta_j$ is reduced to a $y_{kj}$ expression.
So any expression written in $\theta_j$ variables can be transformed into an expression in $y_{kj}$ variables.
