I just found this related question in here Q1.

Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary matrix $\mathbf{B}$,

Is there any closed-form expression of the eigendecomposition of $\mathbf{A} \circ \mathbf{B}$?

$\circ$ denotes the Hadamard product.

**Edition 29/05/2015**

Since considering an arbitrary matrix $\mathbf{B}$ does not lead to a closed-form expression, let us consider that all elements of $\mathbf{B}$ can be defined such as $[\mathbf{B}]_{m,n}= e^{i\theta_{m,n}}$ where $\theta_{m,n} \in [0, 2\pi]$ for $m = 1, \ldots , M$ $n = 1, \ldots , N$. Of course, $\mathbf{A}$ has the same dimensions of $\mathbf{B}$.

Thank you in advance