I have a symmetric, real, and positive matrix $\bf{A}$, for which I have computed an eigendecomposition $\bf{S \Lambda S}^\intercal$. However, I know that the matrix was generated by a symmetric tri-factorization -- i.e., $\bf{A} = \bf{PFP}^\intercal$ for some positive $n \times k$ matrix $\bf{P}$ and a (symmetric) positive $k \times k$ matrix $\bf{F}$.
I would like to recover $\bf{P,F}$ somehow from the eigendecomposition. (I understand there is symmetry up to permutation, but that's fine for my use case.) I'm stuck trying to write down the correct system of equations by which I can do this. So far, I have:
$$\sum_{i} \lambda_{i} \bf{S}_i\bf{S}_i^\intercal = \sum_{ij} F_{ij}P_{i}P_j^\intercal$$
Is there an explicit construction that will change the basis for $\bf{\Lambda}$ as I want? Alternatively, is there an efficient way to compute the tri-factorization that doesn't involve eigendecomposition as an intermediate step?
I can add more details if needed. Thanks!