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vote
Is it possible to find a 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio?
Yes. This can be done for example by placing the desired eigenvalues along the diagonal
$$
\begin{bmatrix}
1 & 0 \\
0 & 2
\end{bmatrix}
$$
since all real diagonal matrices are Hermitian. More ...
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