Relationship of eigenvectors of $A$ and $B$, where $B=MAM$ and $M$ is diagonal.

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Write $A$ and $B$ as two $n\times n$ symmetric matrices, where $B=MAM$ and $M$ is diagonal. Let $V_A$ and $V_B$ be the eigenvector matrices of $A$ and $B$. Is $V_A V_B'$ a function of $M$?