I'm trying to understand the lecture notes comparison theorems in Riemannian geometry of Eschenburg and have a problem about the proof of Rauch I. Let $E$ be an euclidean vector space and $S(E)$ the space of its self adjoint endomorphisms with a partial ordering by putting $ A \leq B $ if $ \langle Ax,x\rangle \leq \langle Bx,x\rangle , \forall x \in E$.


If $A\_{i}: [0, t\_{i}) ,\ i = 1, 2 $ are solutions of the Riccati equations $ A\_{i} ' + A^2 \_{i} + R\_{i} = 0 $ with $ \lambda\_{-} (R\_{1} )\geq \lambda\_{+} (R\_{2} ) $, $ \lambda\_{-} $ the lowest eigenvalue and $ \lambda\_{+} $ the highest eigenvalue of the self adjoint operator $ R\_{i}$, such that $ U := A\_{2} -A\_{1} $ has a
continuous extension to $ 0$ with $U(0) \geq 0 $. Why does it hold that $ \lambda\_{+} (A\_{1} (t)) \leq \lambda\_{-} (A\_{2}(t)) $ on $ (0, t\_{0} ) $?


Thanks a lot.



comparison of eigenvalue of solution of Riccati equation
A classical proof can be found here:


*H. L. Royden*, MR 948079 [**Comparison theorems for the matrix Riccati equation**](http://dx.doi.org/10.1002/cpa.3160410512), *Comm. Pure Appl. Math.* **41** (1988), no. 5, 739--746.


Actually, more general version of this statement holds for more general version of the matrix Riccati equation. A self-contained and perhaps shorter proof, including your case, can be found in Appendix A (Thms. 38 and 40) of


*D. Barilari, L. Rizzi*, MR3491778 [**Comparison theorems for conjugate points in sub-Riemannian
 geometry**](http://dx.doi.org/10.1051/cocv/2015013), *ESAIM Control Optim. Calc. Var.* **22** (2016), no. 2, 439--472.


