Consider two vector spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, where $0<n<m<\infty$. Now, I'd like to define matrices $A\in\mathbb{C}^{n\times n}$ and $B\in\mathbb{C}^{m\times m}$ in the following way.
With $A_i\in\mathbb{C}^{n\times n}$ and $B_i\in\mathbb{C}^{m\times m}$, decide a common set of coefficients $s_i$ to sum up \begin{equation} A=\sum_{i} s_i A_i ~~\mathrm{and}~~ B=\sum_{i} s_i B_i, \end{equation} where the commutation relations are exactly the same, i.e. \begin{equation} [A_i,A_j]=\sum_{k} p_k^{(i,j)} A_k ~~\mathrm{and}~~ [B_i,B_j]=\sum_{k} p_k^{(i,j)} B_k \end{equation} with the same set of coefficients $p_k^{(i,j)}$. For simplicity, let's also assume all matrices here are Hermitian.
With this setup, can we say anything about the relations between the two eigen spectrum between $A$ and $B$? Since $n<m$, it intuitively seems to me that the set of all eigenvalues for $A$, which we will denote $\Lambda(A)$, should be a subset of $B$'s.
Can we actually say $\Lambda(A)\subseteq\Lambda(B)$ ? Can we even go further and show $\Lambda(A)=\Lambda(B)$? Or are there easy counterexamples? Maybe we would need more restriction to have these statements to be true, but in that case what are they? Does anyone know a good reference in general for these kinds of things?
I have a physics back ground and my explanation may not be clear. I'm happy to clarify in that case. Thank you!
----For those who know QM----
My question comes from the following physics intuition: When you have two spin-1/2s, they can form a total spin 1. Also, the summation of Pauli terms like $\sum_i Z_i$ and $\sum X_i$ for all spins $i=1,2,\ldots,k$ gives the same commutation relation as a spin-$k/2$ angular momentum operators like $J_z$ and $J_x$. This makes it seem like you would have essentially the same Hamiltonian when you write one with $J_z$ etc versus with Pauli sums like $\sum Z_i$, but they act on different Hilbert spaces. So, up to differences in the degeneracy, will their eigen spectrum match?