How to rewrite $R^TAR = V$ to find $A$

Question

How to write the following equation to find $V$, the eigenvalue matrix. Note that $R$ is the orthogonal eigenvector matrix, and $A$ is one of the symmetric matrices.

Answer 1

I assume that we have $R^TAR = V$, and $R$ is an orthogonal matrix. It follows that $$ R^TAR = V \implies\\ R(R^TAR) = RV \implies\ \\ AR = RV \implies\\ (AR)R^T = (RV)R^T \implies\\ A = RVR^T. $$