I am currently learning about determinantal point processes, mainly via this survey by Kulesza and Taskar. The likelihood kernel $L$, a $N\times N$ symmetric positive semidefinite matrix, of such a point process is a Gram matrix, given by $$ L = B^TB$$ where the columns of the $D \times N$ matrix $B$ are given by $$ B_i = q_i \phi_i, \quad q_i \in \mathbb{R}^+, \phi_i \in \mathbb{R}^D, \|\phi_i\| = 1$$ for all $i = 1, \dots, N$.
My question is this: Given an eigendecomposition $L = \sum_{n=1}^N \lambda_n v_n v_n^T$ of the likelihood kernel, can I write the eigenvalues and eigenvectors in terms of the components $q$ and $\phi$ of the Gram matrix?
For some context, I find the representation of the point process in terms of these components quite intuitive, as they represent a measure of quality and diversity, respectively. I would like to understand the effect each component has during sampling, which relies on the eigendecomposition.
Edit: After considering this some more, I am inclined to think this is not enough information to represent the eigenvalues and eigenvectors in terms of $q$ and $\phi$. Can anyone confirm this? If not, some ideas on how to proceed would also be helpful.
