Another matrix property question: eigenvalues of a matrix

Question

Let $a = (a_1, \cdots, a_n) \in \mathbb{R}^n$ and $A = \text{diag}(a) \in \mathbb{R}^{n\times n}$. Furthermore, let $B = (b_{ij})_{ij}\in \mathbb{R}^{n\times n}$ be positive-definite symmetric. Suppose I know the eigenvalues of $B$ (let's say $\lambda_1, \cdots, \lambda_n$), and clearly the eigenvalues of $A$ are $a_1, \cdots, a_n$. Are there any expressions/approximations/bounds for the eigenvalues of \begin{align*} M \overset{\text{def}}{=} A B A = aa^\intercal \odot B = (a_i a_j b_{ij})_{ij} \end{align*} where $\odot$ represents Hadamard product. I've played around with all of the three above representations to no avail.