I have a physics inspired problem which I am sure has been solved, but I am lacking the correct terminology for it - perhaps someone can point me in the right direction?
Take two finite-dimensional hermitian operators, $A$ and $B$ of equal dimension $N$ acting on $\mathbb{R}^{N}$. I am interested in when the eigenspace spanned by a subset of eigenstates of $A$ neatly decomposes into eigenspaces of $B$. Thus, take a collection of eigenstates of $A$, $\{|a_n\rangle\}_{n\in M}$ with $M\leq N$, and consider the space $\mathcal{M} = \mathrm{Span}\{|a_n\rangle\}_{n\in M}$. Assume that there are no degenerate eigenvalues. Any eigenstate of $B$ can then be uniquely decomposed as $|b_i\rangle = \alpha_i|\psi_i\rangle+\beta_i|\varphi_i\rangle$ with $|\psi_i\rangle \in \mathcal{M}$ and $|\varphi_i\rangle \in \mathcal{M}^{\perp}$.
Are there any general conditions on $A$ and $B$ which will guarantee that for all $|b_i\rangle$, either $\alpha_i$ is $0$ or $\beta_i$ is $0$, i.e. that each eigenvector of $|b_i\rangle$ exclusively lives in either $\mathcal{M}$ or $\mathcal{M}^{\perp}$?
For $M=N$, this property is trivially satisfied as $\mathcal{M}^{\perp} = \emptyset$. I believe that for $M=1$, this amounts to requiring that $A$ and $B$ commute. Can one say anything generic beyond these two cases?