Project 1, hints

When setting up the algo it is useful to note that the different operations on the matrix (here as a $4\times 4$ case with diagonals $d_i$ and off-diagonals $e_i$) give is an extremely simple algorithm, namely $$\begin{bmatrix} d_1 & e_1 & 0 & 0 \\ e_2 & d_2 & e_2 & 0 \\ 0 & e_3 & d_3 & e_3 \\ 0 & 0 & e_4 & d_4 \end{bmatrix}\rightarrow \begin{bmatrix} d_1 & e_1 & 0 & 0 \\ 0 & \tilde{d}_2 & e_2 & 0 \\ 0 & e_3 & d_3 & e_3 \\ 0 & 0 & e_4 & d_4 \end{bmatrix}\rightarrow \begin{bmatrix} d_1 & e_1 & 0 & 0 \\ 0 & \tilde{d}_2 & e_2 & 0 \\ 0 & 0 & \tilde{d}_3 & e_3 \\ 0 & 0 & e_4 & d_4 \end{bmatrix}$$ and finally $$\begin{bmatrix} d_1 & e_1 & 0 & 0 \\ 0 & \tilde{d}_2 & e_2 & 0 \\ 0 & 0 & \tilde{d}_3 & e_3 \\ 0 & 0 & 0 & \tilde{d}_4 \end{bmatrix}$$

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