as the title says, I look for a way to calculate the 2x2 matrix $A$ if the 2x2 matrices $B$ and $C$ are known with the equation $A$$B$$A$=$C$
Background: I am at my master thesis in electrical engineering and I have to cancel the influence of connectors to a measurement. The influence of the connector is described by a 2x2 matrix(S-parameter) $A$, which I don't know. I know however the S-Parameter of the DUT $B$ and the resulting S-Parameters $C$ by measurement.
Excluding pathological cases for $\bf A$ and $\bf B$, then you can follow these steps $$ \eqalign{ & {\bf A}\;{\bf B}\;{\bf A} = {\bf C} \cr & {\bf A}\;{\bf B}\;{\bf A}\;{\bf B} = \left( {{\bf A}\;{\bf B}} \right)^{\,{\bf 2}} \; = {\bf C}\;{\bf B} \cr & {\bf A}\;{\bf B}\; = \left( {{\bf C}\;{\bf B}} \right)^{\,1/{\bf 2}} \cr & {\bf A} = \left( {{\bf C}\;{\bf B}} \right)^{\,1/{\bf 2}} \;{\bf B}^{\, - \,1} \; \cr} $$
Concerning the square root, pay attention to fact that there might be more than one (or none real): have a look to this Wikipedia article.
One point that might help is to diagonalize, or put in Jordan form, the matrix $\bf{CB}$.