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.