Hubbard Homework.

Matrix for two sites and two electrons:$$ \begin{matrix}
U, &\gamma &-\gamma &0 &0 &0
\\ \gamma &0 &0 &\gamma &0 &0
\\ -\gamma &0 &0 &-\gamma &0 &0
\\0 &\gamma &-\gamma &U &0 &0
\\0 &0 &0 &0 &0 &0
\\0 &0 &0 &0 &0 &0 
\\ \end{matrix} $$

1. a) Find the eigenvalues and eigenvectors of this matrix.
b) What are the 4 eigenvectors that do not have U in their eigenvalue. Which 3 eigenvectors are particularly related? What is the nature of their relationship? (Think about spin. Refer back to the definitions of the vectors of the initial basis states.)