Transforming back and forth between reference frames using orthogonal transformation matrices

Question

The transformation of a covariance matrix $C$ from reference frame 1 to reference frame 2 is described as

\begin{equation} C_2 = R_{12}C_1R_{12}^T \end{equation}

using the (orthogonal) transformation matrix $R_{12}$, where the superscript $T$ denotes the transpose. In this case, I know what $C_2$ is and I want to find $C_1$. How do I use these transformation matrices in order to get $C_1$?

Answer 1

Isolate $C_1$ by multiplying by the appropriate inverses on both sides: $$C_1 = R_{12}^{-1}C_2R_{12}^{-T}$$ and then, since $R_{12}$ is orthogonal so that $R_{12}^{-1}=R_{12}^T$, $$C_1 = R_{12}^TC_2R_{12}.$$