<s>[INST] Finding common eigenvectors for two commuting hermitian matrices Let $A = \begin{bmatrix}
1 &0  &0 \\ 
 0&  0& 0\\ 
 0&0  &1 
\end{bmatrix}$ and  $B = \begin{bmatrix}
0 &0  &1 \\ 
 0&  1& 0\\ 
 1&0  &0 
\end{bmatrix}$ the representation of two hermitian operators in a $(\phi_{1},\phi_{2},\phi_{3})$ basis.  Find a common basis of eigenvectors of the two operators...
So... is easily shown that both matrices commute and are hermitian, the corresponding eigenvalues and eigenvectors are:

*

*For $A$: $a_1 = 0$  with corresponding $\begin{bmatrix}
0\\ 
1\\
0 
\end{bmatrix}$ Eigenvector, $a_2 = 1$ with corresponding $\begin{bmatrix}
1\\ 
0\\
0 
\end{bmatrix} , \begin{bmatrix}
0\\ 
0\\
1 
\end{bmatrix}$ Eigenvectors


*For $B$: $b_1 = 1$ with Eigenvectors $\begin{bmatrix}
1\\ 
0\\
1 
\end{bmatrix}, \begin{bmatrix}
0\\ 
1\\
0 
\end{bmatrix}$ , $b_2 = -1 $ with corresponding $\begin{bmatrix}
1\\ 
0\\
-1 
\end{bmatrix}$ Eigenvector
How can I find a common set of Eigenvectors?
 [/INST] Just use the matrix of the eigenvectors of B:
$$
U = \left( \begin{matrix}
1&1 &0 \\ 
0& 0&1\\ 
1& -1&0
\end{matrix}  \right)
$$
With this matrix, you find that:
$$
U^{-1}AU = \left( \begin{matrix}
1&0&0 \\ 
0& 1&0\\ 
0& 0& 0
\end{matrix}  \right)$$
and $$
U^{-1}BU = \left( \begin{matrix}
1&0&0 \\ 
0& -1&0\\ 
0& 0& 1
\end{matrix}  \right),
$$
which are both diagonal matrices.
</s>