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 $$ \hat{Q}=\left[\hat{q}_1\hat{q}_2\dots\hat{q}_n\right]. $$