Reachability of eigenvalues for additive matrix equation

Suppose I am considering the following matrix-valued equation:

\begin{align} A = B + C,\\ \text{where } A,B,C \in \mathbb{R}^{n\times n} \end{align}

My aim is:

Given $$B$$, I want to find a particular $$C$$ such that I can ensure that $$A$$ obtains a particular set of eigenvalues.

Initially $$A$$ and $$B$$ do not share any common eigenvalues.

My question is:

Is it possible to "reach" every single set of eigenvalues for $$A$$, from $$B$$, using $$C$$? If $$A$$ is to remain symmetric, positive-definite is it possible to find some minimal restrictions on this additive $$C$$ matrix such that this is possible (e.g. $$C$$ only needs to be diagonal? Or $$C$$ only needs to be rank 1).

edit:

I was wondering about two possible constraints for $$C$$:

(1) Is it possible for $$C$$ to be purely diagonal to ensure that all eigenvalues are reachable. Or,

(2) If $$C$$ is rank 1, is it possible to reach all eigenvalues.

• Rank $1$ is certainly not enough, for many reasons. For example, if the rank of $B$ was $\le n-2$, then $B+C$ will still be singular. – Christian Remling Jan 29 at 16:14