1
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\...
Only top voted, non community-wiki answers of a minimum length are eligible