We have thus $$ \hat{A}\hat{q}_k=\beta_{k-1}\hat{q}_{k-1}+\alpha_k\hat{q}_k+\beta_k\hat{q}_{k+1}, $$ with \( \beta_0\hat{q}_0=0 \) for \( k=1:n-1 \). Remember that the vectors \( \hat{q}_k \) are orthornormal and this implies $$ \alpha_k=\hat{q}_k^T\hat{A}\hat{q}_k, $$ and these vectors are called Lanczos vectors.