Is it possible to find a 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio?

Question

Is it possible to find 2x2 Hermitian matrix whose eigenvalues have 1:2 ratio and if it is how is it done?

Answer 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

$$ \sum_k\lambda_k|k\rangle\langle k| $$

is Hermitian.