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Are there any restrictions on the possible spectrum of the sum of two arbitrary matrices with given spectra other than the trace identity? In other words:

Let $\alpha, \beta, \gamma$ be $n$-tuples (nonordered) of complex numbers such that there exist matrices $A,B$ and $C$ respectively, such that $\alpha$ is the spectrum of $A$, $\beta$ is the spectrum of $B$ and $\gamma$ is the spectrum of $C$. Is the set of all possible triples of $\alpha, \beta, \gamma$ is a subset of $\mathbb{R}^{3n}/permutations$ defined by a single equation $\sum \alpha_i+\sum \beta_i=\sum \gamma_i$?

For Hermitian matrices (as well as some other special classes of matrices) the complete answer is well known by Klyachko-Knutson-Tao-... I was wondering whether the structure of the set of triples of possible spectra is trivial or not known or out of reach for all matrices. I don't know the answer even for the case of $2\times 2$ matrices.

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  • $\begingroup$ For $2\times 2$ matrices the answer is yes: the only constraint on the eigenvalues of $C$ is a constraint on the sum $\gamma_1+\gamma_2=\alpha_1+\alpha_2+\beta_1+\beta_2$; the product $\gamma_1\gamma_2=(\alpha_1+\beta_1)(\alpha_2+\beta_2)+\Delta$, where $\Delta$ can take any value independently of $\alpha_1,\alpha_2,\beta_1,\beta_2$, so the product is unconstrained. $\endgroup$
    – Carlo Beenakker
    Commented Jun 28, 2020 at 6:54

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For 2$\times$2 matrices, it suffices the consider the case $$A=\left(\begin{matrix}\alpha_1&0\\ x&\alpha_2\end{matrix}\right),$$ $$B=\left(\begin{matrix}\beta_1&y\\ 0&\beta_2\end{matrix}\right).$$

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