Our Householder transformation has given us a tridiagonal matrix. We discuss here how one can use Householder's iterative procedure to obtain the eigenvalues. Let us specialize to a $4\times 4$ matrix. The tridiagonal matrix takes the form $$\mathbf{A} = \left( \begin{array}{cccc} d_{1} & e_{1} & 0 & 0 \\ e_{1} & d_{2} & e_{2} & 0 \\ 0 & e_{2} & d_{3} & e_{3} \\ 0 & 0 & e_{3} & d_{4} \end{array} \right).$$ As a first observation, if any of the elements $e_{i}$ are zero the matrix can be separated into smaller pieces before diagonalization. Specifically, if $e_{1} = 0$ then $d_{1}$ is an eigenvalue.