Consider the following nonlinear matrix equation:

$B=PX^{−1}AX$

where $B$ and $P$ are a $1\times n$ row vector and $A$ is a $n\times n$ matrix which are all strictly positive, and $X=diag(x_1,...,x_n)$ is a diagonal matrix with $x_i>0$ for all $i$'s. $X$ is the unknown while the rest are all taken as given.

I'm looking for the general solution for $X$ and for its existence and uniqueness conditions. But it seems that the general solution does not exist. If so, how can we formally prove that?

I read this post but it had no answer for me.