In my research on linear algebra and its applications, I have come across the following problem which has stumped me:

Let $ A $ be a positive definite matrix and let $ D $ be a positive diagonal matrix with entries on the main diagonal: $ d_1,d_2,...,d_n $, both $ A $ and $ D $ have the same dimension $ n \times n $. I was interested in understanding how the eigenvalues of the sum $ A + ADA $ qualitatively behave with respect to the eigenvalues of $ A $ and the entries $ d_1,...,d_n $.

I thought since this sum has a special form, one could hopefully say a bit more analytically than by exclusively using the fact that the two summands are Hermitian. I thought about using some other techniques, perhaps expressing the product of $A$ and $ D $ as a polynomial of $ A $, but unfortunately, I am stuck. I certainly appreciate all help on this.