Given a full-rank matrix $X$, and assume that the eigen-decomposition of $X$ is known as $X=V \cdot D \cdot V^{-1}$, where $D$ is a diagonal matrix.
Now let $C$ be a full-rank diagonal matrix, now I want to calucate the eigen-decomposition of $C \cdot X$, that is to find a matrix $V_c$ and a diagonal matrix $D_c$ such that $C \cdot X =V_c \cdot D_c \cdot V_c^{-1}$. Since the eigen-decomposition of $X$ is known, how can we obtain $V_c$ and $D_c$ from $V$ and $D$, respectively? Thanks!