Thus, let us introduce a transformation $\mathbf{S_{1}}$ which operates like $$\mathbf{S_{1}} = \left( \begin{array}{cccc} \cos \theta & 0 & 0 & \sin \theta\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \cos \theta & 0 & 0 & \cos \theta \end{array} \right)$$

Then the similarity transformation $$\mathbf{S_{1}^{T} A S_{1}} = \mathbf{A'} = \left( \begin{array}{cccc} d'_{1} & e'_{1} & 0 & 0 \\ e'_{1} & d_{2} & e_{2} & 0 \\ 0 & e_{2} & d_{3} & e'{3} \\ 0 & 0 & e'_{3} & d'_{4} \end{array} \right)$$ produces a matrix where the primed elements in $\mathbf{A'}$ have been changed by the transformation whereas the unprimed elements are unchanged. If we now choose $\theta$ to give the element $a_{21}^{'} = e^{'}= 0$ then we have the first eigenvalue $= a_{11}^{'} = d_{1}^{'}$. (This is actually what you are doing in project 2!!)