The matrix \( \hat{A} \) is transformed into a tridiagonal form and the last step is to transform it into a diagonal matrix giving the eigenvalues on the diagonal.

The eigenvalues of a matrix can be obtained using the characteristic polynomial $$ P(\lambda) = det(\lambda\mathbf{I}-\mathbf{A})= \prod_{i=1}^{n}\left(\lambda_i-\lambda\right), $$ which rewritten in matrix form reads $$ P(\lambda)= \left( \begin{array}{ccccccc} d_1-\lambda & e_1 & 0 & 0 & \dots &0 & 0 \\ e_1 & d_2-\lambda & e_2 & 0 & \dots &0 &0 \\ 0 & e_2 & d_3-\lambda & e_3 &0 &\dots & 0\\ \dots & \dots & \dots & \dots &\dots &\dots & \dots\\ 0 & \dots & \dots & \dots &\dots &d_{N_{\mathrm{step}}-2}-\lambda & e_{N_{\mathrm{step}}-1}\\ 0 & \dots & \dots & \dots &\dots &e_{N_{\mathrm{step}}-1} & d_{N_{\mathrm{step}}-1}-\lambda \end{array} \right) $$