Let $A$ and $B$ be two positive semidefinite $n \times n$ matrices. Does the following quadratic matrix equation have a solution in the set of real symmetric matrices?
$$XAX=B$$
It's a special case of the Riccati equation. I want just to prove the existence of such real matrix $X$.
Let's assume $A$ is positive definite. Multiplying on left and right by $A^{1/2}$ (the positive definite square root of $A$) the equation becomes $$ (A^{1/2} X A^{1/2})^2 = A^{1/2} B A^{1/2}$$ Now $A^{1/2} B A^{1/2}$ is positive semidefinite, so has a positive semidefinite square root $C$, and we can take $X = A^{-1/2} C A^{-1/2}$ so that $C = A^{1/2} X A^{1/2}$.