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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 generally, for any desired eigenvalues $\lambda_k\in\mathbb{R}$ and any orthonormal basis $|k\rangle$, the matrix of the operator  \sum_k\lambda_k|k\rangle\...