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I have this equation:

$$x_i = \frac{\alpha_i^2}{\sum{\alpha_j}}$$

and transformed it into vector mode this way:

$$x = \frac{\alpha^T \alpha}{1^T \alpha}$$

where x and $\alpha$ are column vectors of size $n$, and 1 is a $n$ size vector of all 1s. Given $x$, how can I calculate $\alpha$ in this situation? One constraint is that all the elements of $\alpha$ have to be between 0 and 1.

Thanks!

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    I think you're missing a vector of 1s in the numerator. As it's currently written, you'll have an $n\times n$ matrix there. – 2017-01-20
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    And if you don't know anything else about $\alpha$, like that it's entries sum to some constant $A$, you won't be able to recover $\alpha$ from $x$. – 2017-01-20
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    @Max I think OP would have an $n \times n$ matrix if they had $\alpha \alpha^{T}$. Right now it's $1 \times 1$, which is still a problem since $x$ should be a vector while the right hand side is a number. – 2017-01-20
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    @Max, you are right, there is an error in my second equation. But why can't I solve the system given by my first equation? – 2017-01-20
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    The solution is $\alpha_i=\sqrt{x_i\sum\alpha_j}$. You need to know something about $\sum\alpha_j$ to solve it. – 2017-01-20

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