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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.

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  • $\begingroup$ 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. $\endgroup$ – Christian Remling Jan 29 at 16:14

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