Alternative way to do this eigenvector problem Just wondering if there is a faster or better way of doing this question.
I have 3 matrices: $A = {1\over2}\hbar\left( \begin{smallmatrix} 0&-i\\ i&0 \end{smallmatrix} \right), B = {1\over2}\hbar\left( \begin{smallmatrix} 0&1\\ 1&0 \end{smallmatrix} \right), C = {1\over2}\hbar\left( \begin{smallmatrix} 1&0\\ 0&-1 \end{smallmatrix} \right), $
Let $D = A^2+B^2+C^2$ and that $v$ is an eigenvector of $C$, (with eigenvalue $-{1\over2}\hbar$).
Question: Show that $v$ is an eigenvector of $D$ with eigenvalue ${1\over2}({1\over2}+1)\hbar^2$.
My take: $D={3\over4}\hbar^2\left( \begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \right)$. $Cv=-{1\over2}\hbar v \implies C^2v={1\over4}\hbar^2 v$. 
Since $C^2={1\over4}\hbar^2\left( \begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \right)=B^2=A^2$, Therefore $Dv = 3\times {1\over4}\hbar^2v$. So we are done.
Though I do get the numerical answer, it is not immediately in the form asked for. Could anyone suggest another way of doing this? Thank you.
 A: The matrices $A$, $B$, $C$ are multiples of the Pauli matrices. Their commutation relations are
$$[B,A]=\mathrm i\hbar C$$
and cyclic permutations thereof. These are the commutation relations of angular momentum operators, or, mathematically speaking, of generators of $SU(2)$, which form a basis of $\mathfrak{su}(2)$, the Lie algebra of $SU(2)$. They provide an irreducible representation of $\mathfrak{su}(2)$ (the defining representation). The Casimir invariant given by the sum of the squares of the generators acts as a multiple of the identity within an irreducible representation, the factor being $j(j+1)$, where $j$ labels the irreducible representations and may be either integer or half-integer. The defining representation has $j=\frac12$, so $j(j+1)=\frac12(\frac12+1)=\frac34$. So not only eigenvectors of $v$ are eigenvectors of $D$; all vectors are eigenvectors of $D$ with the same eigenvalue. (This is only true within an irreducible representation; a reducible representation will generally contain eigenvectors of the squared angular momentum operator corresponding to different values of $j$.)
There's obviously a lot more to be said about all this; I hope I've given you some pointers what to look at if you're interested in the background of this question.
