What is this method of scaling called? Can it be generalised? Consider the problem of finding the values of $\alpha_1, \alpha_2, ..., \alpha_k$, subject to constraints, such that the following equation is satisfied
\begin{equation}
\alpha_1 x_1 + \alpha_2 x_2 + \dots \alpha_k x_k = P
\end{equation}
where all $x_i$'s and $P$ are real numbers.
For example, the constraint of uniform scaling of the $x_i$'s would be setting $\alpha_i = \frac{P}{\sum_i x_i}$.
Now I wish to scale the $x_i$'s such that the ratio between $\alpha$'s is equal to the ratio between the respective $x$'s, that is:
\begin{equation}
\frac{\alpha_m}{\alpha_n} = \frac{x_m}{x_n} 
\end{equation}
for all $m, n$.
A solution is to set:
\begin{equation}
\alpha_i = \frac{x_i P}{x_1^2 + x_2^2 + \dots + x_k^2}
\end{equation}
What is this called? Is there anyway to find this solution quickly without knowing it before hand?
Can it be generalised?
 A: Do you mean something like this?
\begin{eqnarray*}
\frac{1}{\alpha _{j}}P &=&\frac{1}{\alpha _{j}}\{\alpha _{1}x_{1}+\cdots
+\alpha _{k}x_{k}\}=\frac{\alpha _{1}}{\alpha _{j}}x_{1}+\frac{\alpha _{2}}{%
\alpha _{j}}x_{2}+\cdots x_{j}+\cdots \frac{\alpha _{k}}{\alpha _{j}}x_{k} \\
&=&\frac{x_{1}^{2}}{x_{j}}+\frac{x_{2}^{2}}{x_{j}}+\cdots \frac{x_{k}^{2}}{%
x_{j}}=\frac{1}{x_{j}}\sum_{j=1}^{k}x_{j}^{2}\Rightarrow \alpha _{j}=\frac{%
x_{j}}{\sum_{j=1}^{k}x_{j}^{2}}P
\end{eqnarray*}
In the same way
\begin{equation*}
x_{j}=\frac{\alpha _{j}}{\sum_{j=1}^{k}\alpha _{j}^{2}}P
\end{equation*}
You can look upon $\{\alpha _{1},\cdots ,\alpha _{k}\}$ and $\{x_{1},\cdots
,x_{k}\}$ as $k$-dimensional vectors, so
\begin{equation*}
\mathbf{\alpha \cdot x}=P
\end{equation*}
Note that the component of $\mathbf{x}$ orthogonal to $\mathbf{\alpha }$ is
arbitrary and vice versa. With your additional conditions you find that $%
\mathbf{\alpha }$ is directed along $\mathbf{x}$ and for their norms
\begin{equation*}
|\mathbf{x}||\mathbf{\alpha }|=P
\end{equation*}
