Diagonalization and the Hadamard product

Question

Let $B \in \mathbb{C}^{n\times n}$ be unitarily diagonalizable such that $B=V\Lambda V^*$. Let $A=B\circ B$ where $\circ$ accounts for the Hadamard product. Then we can say that $A$ is also unitarily diagonalizable. I need a hint for unitarily diagonalization of the matrix $A$ based on $V$ and $\Lambda$. Can we do that? At least can we get any information about the eigenvalues of $A$ based on eigenvalues of $B$?