I have found an answer to my question on Mathoverflow.net [here][1]. @Raziel gave the reference 

H. L. Royden, MR 948079 Comparison theorems for the matrix Riccati equation, [Comm. Pure Appl. Math. 41 (1988), no. 5, 739--746.][2]

where I can find a comparison theorem for eigenvalues of $S$ but **not** for eigenvalues of $\nabla\nabla \eta_\Sigma$.  

I keep the question and the answer on Mathoverflow since it might be useful for someone else.


  [1]: https://mathoverflow.net/questions/249184/comparison-of-eigenvalue-of-solution-of-riccati-equation?rq=1
  [2]: http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160410512/abstract