Help with solving a parametric matrix equation

Question

I have the following equation:

$B^{-1}E^{-1}(B^{-1}R^{-1}BE)^{\frac{1}{2}}B=Q$

B, E,R and Q are all square, reversible matrices.

I need to find an expression for B.

any ideas?

Answer 1

I assume that your matrices are real. If you want to obtain solutions in $B$, then necessarily, $\det(R)(\det(Q))^2\det(E)=1, \det(E)\det(Q)>0$.

Moreover, when solutions exist, there is an infinity; for example, when $n=2$, in general, the degree of freedom is $1$.

Of course, there does not exist any closed form for the solutions.