Eigenvectors of Hadamard product of a positive definite matrix and a symmetric positive semidefinite matrix

Question

Let $b \in (0,1]$ and $n<\infty$. Consider the following square matrices:

1. Symmetric positive semi-definite matrix $A \in M_{n \times n}(\mathbb{R})$, and
2. $B \in M_{n \times n}(\mathbb{R})$ with $(i,j)^{th}$ entries defined by $B_{i,j} = b^{i-1} \cdot b^{j-1}$.

Since $b >0$, $B$ is positive definite. Let $\odot$ denote the Hadamard product.

Suppose if the eigenspace of $A$ is known. Is there anything we can say about the eigenspace of $B \odot A$?

I have considered the following:

1. By Lemma 3.1 of [1], Hadamard product $B \odot A$ is a principal submatrix of the Kronecker product $B \otimes A$. But I'm not sure whether the projection onto the submatrix would give us any information on the eigenspace.

2. Let $b_0 \in (0,1)$, $b_1 = 1$ and consider the line segment $b(t) = t \cdot b_1 + (1-t) \cdot b_0$. Let $B(t) \in M_{n \times n}(\mathbb{R})$ be the positive definite matrix with $(i,j)^{th}$ entries defined by $B(t)_{i,j} = b(t)^{i-1} \cdot b(t)^{j-1}$. Is there anything we say about the eigenspace of the curve $B(t) \odot A$?

3. Consider diagonal matrix $D \in M_{n \times n}(\mathbb{R})$ with $D_{i,i} = b^{i-1}$ and $0$ otherwise. Then $B \odot A = DAD$. Since $A$ is symmetric and $D$ diagonal, are the eigenvectors of $DAD$ related to those of $A$?

At this point I'm a bit stuck and any help would be very much appreciated. Thanks you very much in advance!

References: [1] Pukelsheim, Friedrich. "On Hsu's model in regression analysis." Statistics: A Journal of Theoretical and Applied Statistics 8.3 (1977): 323-331.