Solving a nonlinear matrix equation I'm trying to find the solution for $X$ of the following nonlinear matrix equation:
$F=PX^{−1}AX+wL$
where $w>0$ is a scalar, $F$, $P$, and $L$ are all a $1\times n$ row vector and $A$ is a $n\times n$ matrix of $a_{ij}$ and $X=diag(x_1,...,x_n)$ is a diagonal matrix. All the matrices and vectors are strictly positive.
$X$ is the unknown while the rest are all taken as given. 
Edit: As shown in Robert Israel's Answer below, the system doesn't seem to have a nice symbolic solution in a general case. Then a new question: How can we formally prove this?
Thanks!
 A: Let B = F-wL and consider the following reformulation/generalization of your problem:
Given A, B and P find diagonal X such that PX-1AX = B
(it is a generalization in the sense that I have not required entry-wise positivity on the variables)  
Let Y = X-1, then we have PYAY-1 = B or equivalently PYA = BY where Y has no zero entries  
PYA is a row vector with each entry linear in the entries of Y, similarly for BY
Thus PYA = yV for some matrix V and BY = yW for some matrix W where y is the row vector of entries of Y
So we get the equation yV = yW or equivalently KyT = 0 where K = (V-W)T
So we need to find the matrix K which can easily be found from A, B and P, and then find the null space of K. Once we have the null space we need a vector in the null space without any zero entries. If you also require that X only have positive entries (I'm not sure if you meant X to be included when you say "All the matrices and vectors are strictly positive") then you instead want to find a point in the null space where all entries are positive.
Side note:
I'm a bit unclear on what needs to be positive. Consider the matrix X, it is diagonal and thus contains zero entries however all entries of matrices must be strictly positive, so this can't be. If you could clarify here that would be helpful.
In any case, I do suspect this problem can be reduced to intersecting a subspace (in this case the null space of a specific matrix) with the set of vectors with non zero entries, or in the second case intersecting a subspace with the set of vectors with all positive entries. The second case is a linear program (see here: https://en.wikipedia.org/wiki/Linear_programming). The first case however is more complex.
A: According to the Fred's post, the problem reduces to the equation in the unknown $Y$ (a positive diagonal matrix) $(*)$ $pYA=bY$ where the vectors $p,b$ and the matrix $A$ are known and positive.
$(*)$  is equivalent to $(p\otimes A^T-b\otimes I_n)Y=0$ (if we stack row by row the matrix into a vector) cf. 
https://en.wikipedia.org/wiki/Kronecker_product
In fact, $(*)$ is the following linear homogeneous system of $n$ equations in $n$ unknowns $(y_i)$ that are non-zero
$(U-diag(b))[y_1,\cdots,y_n]^T=0$  where $U=[u_{i,j}]$ is the $n\times n$ matrix defined by $u_{i,j}=p_ja_{j,i}$.
As noted by Robert Israel, there is a necessary condition of existence; it can be written
$\det(U-diag(b))=0$.
In a second step, we search vectors (if they exist) in $\ker(U-diag(b))$ s.t. $y_1=1$ and, for every $i>1,y_i>0$. Standard softwares do easily the job (with explicit $b,p,A$).
