KrylovUnterräumen

Krylov subspaces are fundamental concepts in numerical linear algebra, particularly in the iterative solution of large sparse linear systems and eigenvalue problems. A Krylov subspace is the vector subspace spanned by the first k vectors in the sequence v, Av, A^2v, ..., A^(k-1)v, where A is a given matrix and v is a non-zero vector, often called the starting vector. These subspaces are denoted as K_k(A, v).

The significance of Krylov subspaces lies in the fact that many popular iterative methods, such as the conjugate gradient method for symmetric positive-definite systems and the Arnoldi/Lanczos methods for general systems and eigenvalue problems, implicitly or explicitly construct bases for these subspaces. The algorithms aim to find an approximate solution to Ax=b or an approximate eigenvalue within a successively larger Krylov subspace.

The dimension of K_k(A, v) is at most k. The dimension becomes stationary when the vectors v, Av, ..., A^(k-1)v become linearly dependent. This linear dependency is directly related to the minimal polynomial of A with respect to v. If the dimension of K_k(A, v) is less than k, then the basis vectors can be expressed as linear combinations of the previous ones, and the algorithms can exploit this structure to become more efficient. The properties of the matrix A, such as being symmetric, self-adjoint, or having specific spectral properties, further influence the structure and utility of the Krylov subspaces.