I'm interested in the following problem:
Problem: Given two matrices $A,B \in \mathbb{C}^{n \times n}$, both with $n$ distinct eigenvalues, find a matrix $C \in \mathbb{C}^{n \times n}$ with minimum Frobenius norm $\lVert C \rVert_F$ such that $A$ is similar to $B+C$.
Variant: $A$, $B$, and $C$ are all restricted to be diagonal matrices. In this case $C$ can be found with a solution to the linear sum assignment problem, taking the cost matrix to be the pairwise squared distances from the diagonal entries of $A$ to the diagonal entries of $B$. The solution gives a permutation matrix encoding which entries (eigenvalues) of $B$ are transformed to which entries (eigenvalues) of $A$.
Motivation: I came across this problem while studying a multivariate Ornstein-Uhlenbeck Process state space model:
$$\text{d}\mathbf{y} = A \mathbf{y} \; \text{d}t + \text{d}W$$ $$\mathbf{x}(t) = F\mathbf{y}(t)$$
where $A,F \in \mathbb{C}^{n \times n}$, $\mathbf{x}, \mathbf{y} \in \mathbb{C}^n$, $\text{d}W$ is a complex multivariate Wiener process, and only $\mathbf{x}(t)$ is observed for all times $t \in \mathbb{R}$. If $F$ is unrestricted, then $A$ is identifiable only up to similarity. In the problem setup I'm considering, I know $A = B+C$ for known $B$ and small $C$. Then a solution to the problem above could be used to find an appropriate $C$.