Consider three hermitian matrices $L$, $K$ and $X$ such that $[L,K]=0$, $[X,K]=0$, but $[L,X] \not =0$, What can we say about the characteristics of each of the matrices?
I understand that since the commutators of the pairs are 0, this implies that they share eigenvectors(but not necessarily eigenvalues), but I am not sure of the general characteristics that $L,K,X$ must satisfy in order to produce such a relationship. My idea was to use a spectral decomposition for each, but after writing each of the matrices in the form $L=U \Lambda U^{-1}$ I got a bit stuck. Since these matrices share eigenvectors, would they all have the same $U's$? Or? I understand that the $\Lambda's$ for each would be different, but im struggling to see how to find what he characteristics of each must be.
Thanks
The eigenspaces for eigenvalues of $K$ are invariant under $L$ and $X$, but the restrictions of $L$ and $X$ to these eigenspaces are essentially arbitrary.