 Lanczos algorithm

For a real symmetric matrix $A$, the leading block $Q_k^{T}A{Q}_k$ of the Arnoldi decomposition is both symmetric and an upper Hessenberg matrix, hence a tridiagonal matrix. The resulting three-term recurrence is the Lanczos algorithm. If $A$ is a symmetric positive-definite matrix, its compression is too, since $y^T{Q}_k^{T}A{Q}_{k}y=(Q_{k}y{)}^{T}A(Q_{k}y)>0$ for nonzero $y$.

The Lanczos algorithm is an iterative numerical method used for solving large eigenvalue problems, particularly those that arise in the context of large sparse matrices. It was developed by Cornelius Lanczos in the 1950s as a way to find a few eigenvalues and corresponding eigenvectors of a Hermitian (or symmetric) matrix.