Approximating eigenvalue and eigenvector pair with Lanczos iteration

Question

Using the Lanczos algorithm for some matrix $A$, I obtained a tridiagonal matrix $T$ and an orthogonal matrix $Q$ such that

$$Q^T A Q = T$$

I also approximated some eigenpair of $T, \ (\lambda_T, \mathbf{v}_T)$

How can I use this eigenpair to approximate its corresponding eigenpair of $A$?

Answer 1

Let $w = Qv_T$. Then $$ \|Aw-\lambda_T w\| = \|AQv_T-\lambda_T Qv_T\| = \|QQ^tAQv_T-Q\lambda_Tv_T\| = \|Tv_T-\lambda_Tv_T\|. $$ Hence $(Qv_T, \lambda_T)$ is an approximate eigenpair for $A$.