Solving a linear matrix equation Could you please help me solve this problem?
Given a $n$ by $m$ matrix $Q$ whose columns are orthonormal ($rank(Q)=m$) and a $n$ by $n$ symmetric definite positive matrix $X$ (the unknown) and another $n$ by $n$ symmetric definite positive matrix $B$.
We want to solve the following problem:
$$QQ^{t}XQQ^t=B$$
We put $A=QQ^t$, the problem is then formulated as: $AXA=B$ in this case $det(A)=0$ 
One manipulation that could be done is:
$$Q^tQQ^tXQQ^tQ=Q^tBQ=Q^tXQ$$ since $$Q^tQ=I$$ (Vectors are orthonormal)
Now I'm stuck with the following equation, is there any ideas to solve it to find X? 
$$Q^tXQ=Q^tBQ$$
or 
$$Q^t(X-B)Q=Q^tMQ=0$$
Thank you in advance.

If I understand the statement of the problem correctly, $A=Q Q^t$ is an orthonormal projection onto an $m$ dimensional subspace of ${\Bbb R}^n$: $A^2=A=A^t$.
You have a solution iff the image of $B$ is in the image of $A$. In this case $ABA=B$ as well (when $B$ is symmetric) so a solution is simply $X=B$. If $B\neq ABA$ then your problem has no solution.
