How are Hermitian operators derived

Question

The von Neumann-Dirac Theory postulates that physical observables are represented by Hermitian operators. For example, let's assume the physical observable I am measuring is the spin of an electron, and my orthonormal basis is composed of the eigenvectors $$ |\uparrow\rangle=\left(\begin{array}{c} 1 \\ 0 \\ \end{array}\right) $$ and $$ |\downarrow\rangle = \left(\begin{array}{c} 0 \\ 1 \\ \end{array}\right) . $$

The Hermitian operator for these eigenvectors is $$ \hat{A} = \left( \begin{array}{c} 1 & 0 \\ 0 & -1 \\ \end{array}\right) , $$ because each eigenvector corresponds to an eigenvalue. In the case above, $|\uparrow\rangle$ has an eigenvalue of $1$ and $|\downarrow\rangle$ has an eigenvalue of $-1$. However, if I wasn't given the Hermitian operator $\hat{A}$ how would I derive it? Would it even be logical to derive it only knowing the eigenvectors?

Answer 1

Since a Hermitian matrix can be diagonalized as $$ A = U^{\dagger} D U , $$ where $U$ is composed of the eigenvectors and $D$ is a diagonal matrix with the eigenvalues on the diagonal, one can use it to compose $A$. For the two-dimensional case it would be represented by $$ A = |a\rangle \lambda_a \langle a| + |b\rangle \lambda_b \langle b| , $$ where $|a\rangle$ and $|b\rangle$ are the two eigenvectors. If you don't know the eigenvalues, it is reasonable to pick $1$ and $-1$ as their eigenvalues, because they are real valued (as required for a Hermitian operator), and would give convenient measurement values.