Lanczos process

• Much stronger results are available for symmetric matrices.
• The algorithm is called Lanczos.
• Lanczos: $T=Q^{T}AQ$
• Symmetric + upper Hessenberg = tri-diagonal

$$T=\left(\begin{array}{ccccc}{\alpha }_1& {\beta }_1& \cdots & & 0\\ {\beta }_1& {\alpha }_2& \cdots & & \\ ⋮& & \ddots & & ⋮\\ & & & \ddots & {\beta }_{n-1}\\ 0& & \cdots & {\beta }_{n-1}& {\alpha }_n\end{array}\right)$$

As a result, the steps in the Lanczos algorithm are simplified compared to Arnoldi:

1. ${\alpha }_k={\mathbf{q}}_k^{T}A{\mathbf{q}}_k$
2. ${\mathbf{r}}_k=A{\mathbf{q}}_k-{\beta }_{k-1}{\mathbf{q}}_{k-1}-{\alpha }_k{\mathbf{q}}_k$; ${\beta }_{k-1}$ is from the previous iteration.
3. ${\beta }_k{\mathbf{q}}_{k+1}={\mathbf{r}}_k$

See Algorithm for the Arnoldi process, Arnoldi process.