We notice the sub-blocks which get repeated $$ \begin{bmatrix} d_1 & e_1 & 0 & 0 \\ 0 & \tilde{d}_2 & e_2 & 0 \\ 0 & 0 & \tilde{d}_3 & e_3 \\ 0 & 0 & 0 & \tilde{d}_4 \end{bmatrix} $$ The matrices we often end up with in rewriting for for example partial differential equations, have the feature that all leading principal submatrices are non-singular.