Basic features with a real symmetric matrix (and normally huge $n> 10^6$ and sparse) $\hat{A}$ of dimension $n\times n$:

• Lanczos' algorithm generates a sequence of real tridiagonal matrices $T_k$ of dimension $k\times k$ with $k\le n$, with the property that the extremal eigenvalues of $T_k$ are progressively better estimates of $\hat{A}$' extremal eigenvalues.* The method converges to the extremal eigenvalues.
• The similarity transformation is

$$\hat{T}= \hat{Q}^{T}\hat{A}\hat{Q},$$ with the first vector $\hat{Q}\hat{e}_1=\hat{q}_1$.

We are going to solve iteratively $$\hat{T}= \hat{Q}^{T}\hat{A}\hat{Q},$$ with the first vector $\hat{Q}\hat{e}_1=\hat{q}_1$. We can write out the matrix $\hat{Q}$ in terms of its column vectors $$\hat{Q}=\left[\hat{q}_1\hat{q}_2\dots\hat{q}_n\right].$$