Let $\mathbf{x}$ be an $n$-dimensional real vector (unknown).
Let $\mathbf{A}$ be an $n \times n$ real matrix (possibly non-symmetric).
Let $\mathbf{b}$ be an $n$-dimensional real vector (known).
Consider the following equation
$\mathbf{b}\;=\;\mathrm{diag}\left(\mathbf{x}\right)\mathbf{A}\mathbf{x}$
where $\mathrm{diag}\left(\mathbf{x}\right)\;=\;\left[\begin{array}{cccc} x_{1} & 0 & \cdots & 0\\ 0 & x_{2} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & x_{n} \end{array}\right]$
Is there a good soul out there that can:
1) Tell me whether there is a closed-form ($\mathbf{x}$) solution to the equation above?
2) If yes, which one is it. If no, how can I find $\mathbf{x}$ numerically (say, in MATLAB). $\mathbf{x}$, $\mathbf{b}$ and $\mathbf{A}$ are real data objects and $n \approx 5000$
Closed-form? I don't think so. I tried a $3 \times 3$ case with random $A$ and $b$. Your equation became three quadratics in $x_1, x_2, x_3$: $$\eqalign{ 85\,{x_{{1}}}^{2}+27\,x_{{2}}x_{{1}}+45\,x_{{1}}x_{{3}}-3 &= 0\cr85\,x_{{2}} x_{{1}}+96\,{x_{{2}}}^{2}+89\,x_{{2}}x_{{3}}+5 &=0\cr-48\,x_{{1}}x_{{3}}+76 \,x_{{2}}x_{{3}}-44\,{x_{{3}}}^{2}-11 &= 0}$$
Solving with Groebner basis methods, $x_3$ must satisfy a rather awful polynomial (irreducible over the rationals) of degree $8$: $$ 85991383888960\,{x_{{3}}}^{8}+119524075710384\,{x_{{3}}}^{6}+ 45323805092604\,{x_{{3}}}^{4}+4288390645605\,{x_{{3}}}^{2}+ 175173708600 = 0 $$ By the way, this has no real roots.
Extrapolating, I wouldn't be surprised if for $n=5000$ the polynomial had degree $2^{5000}$. So there's no way you can find an exact solution (even if you count algebraic numbers as in principle "closed form"). Numerical methods are the only realistic option. And, as this example also shows, don't expect there to always be a real solution.