Given a diagonal matrix $A$ with all entries $A_{i,i}\in(0,1)$ and trace equal to one
$$ A\in \mathbb{R}^{k \times k} $$
and a second matrix
$$ B\in \mathbb{R}^{n \times k},\,\,\,\,\,\,n\geq k,\,\,\,\,\,\, B_{i,j}\in[0,1) $$
I am looking for any kind of theory studying the solutions of the system
$$ c = BAB^t ,\,\,\,c\in \mathbb{R}^{n \times n} $$
for fixed $c$, where the known parameters are the matrix $A$, and the matrix $c$, while $B$ is unknown.
My hope is that something related to the geometric properties of the bilinear forms can help in some way, but also numerical approaches are welcome!
Thanks