This question comes from a colleague working in econometrics. $A$ and $B$ are $n\times n$ real symmetric matrices. If we know the eigenvalues of $A$, $B$ and $A+B$, what meaningful information can we obtain about the eigenvalues of $A^2+B^2$?


I have read this [related question](https://mathoverflow.net/questions/4224/eigenvalues-of-matrix-sums), but here we have the extra information of the eigenvalues of $A+B$.



Eigenvalues of $A^2+B^2$ from those of $A$, $B$ and $A+B$
This is a partial answer, that could evolve in a few hours.


If $\theta+\nu>0$, then you have
$$A^2+B^2\ge(1-\theta)A^2+(1-\nu)B^2+\frac{\theta\nu}{\theta+\nu}(A+B)^2.$$
With Horn's inequalities, you deduce inequalities for the eigenvalues of $A^2+B^2$ in terms of those of $A^2$, $B^2$ and $(A+B)^2$. The latter are the squares of the eigenvalues of $A$, $B$ and $A+B$.


For instance, using Weyl's inequality
$$\lambda\_k(F+G)\ge\lambda\_i(F)+\lambda\_j(G),\qquad k+1=i+j,$$
we obtain
$$\lambda\_m(A^2+B^2)\ge\lambda\_i((1-\theta)A^2)+\lambda\_j((1-\nu)B^2)+\lambda\_k\left(\frac{\theta\nu}{\theta+\nu}(A+B)^2\right),\qquad m+2=i+j+k.$$


