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.