Lanczos Tridiagonalization
Turn a symmetric matrix into a small tridiagonal model along its Krylov basis.
Free theory articleTurn a symmetric matrix into a small tridiagonal model along its Krylov basis.
Concept & interpretation
The Lanczos algorithm computes an orthonormal basis for the Krylov subspace of a real symmetric matrix A. By enforcing orthogonality against previous vectors, it effectively reduces the original matrix to a symmetric tridiagonal form T.
Experiment
In the Lanczos Tridiagonalization module, you can observe the algorithm step-by-step.
- Select Matrix as "Diagonal" and Start as "Uniform".
- Set Iterations to 1. Observe the first alpha and beta values being computed from the first basis vector.
- Change Iterations to 2. The basis dimension increases to 2, and the tridiagonal matrix T grows accordingly.
- Select Start as "Axis". Notice how the algorithm breaks down happily at step 1 because the starting vector is an eigenvector of the matrix. The residual norm drops to 0.