Let be $A, B$ two matrices $n \times n$ with respective eigenvalues $(a_1, \ldots, a_n)$ and $(b_1, \ldots, b_n)$, let be $C = A + B$ and $(c_1, \ldots, c_n) \in \mathbb{R}^n$.
Under which conditions on $A, B$ and $(c_1, \ldots, c_n)$ do we have that $C$ have $(c_1, \ldots, c_n)$ as eigenvalues ?
First, I tried to consider $A, B$ diagonalizable in the same basis, so that I could just say: $\forall i \in [[1, n]], c_i = a_i + b_i$ would suffice.
But I have no idea how to go to get a necessary condition.
Preferably, I am trying to avoid Jordan normal form.
Such questions are research questions. An exercise from Horn and Johnson 2 reveals that unless there are special properties of the matrices involved and more conditions are given, then nothing special happens.
Consider: $A=\begin{bmatrix} 1-\alpha& 1\\ \alpha(1-\alpha)-1& \alpha\end{bmatrix}$ and $B=\begin{bmatrix} 1+\alpha& 1\\ -\alpha(1+\alpha)-1& -\alpha\end{bmatrix}$. The eigenvalues of $A$ and $B$ are independent of $\alpha$, but the spectral radius of $A+B$ is unbounded as $\alpha\to\infty$.
Another way to see my point is: specifying the eigenvalues of $A$, $B$, $C$ without constraints whatsoever is too weak an assumption to get interesting results. And once you do impose constraints, then things go to the shadow realm of research.