KrylovUnterräume

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 linear span of vectors generated by repeatedly applying a matrix A to an initial vector v. Specifically, for a given square matrix A and a non-zero vector v, the k-th Krylov subspace, denoted by $K_k(A, v)$, is defined as the set of all linear combinations of the vectors $\{v, Av, A^2v, \dots, A^{k-1}v\}$. Mathematically, this can be expressed as $K_k(A, v) = \text{span}\{v, Av, A^2v, \dots, A^{k-1}v\}$.

These subspaces are crucial because they often contain good approximate solutions to linear systems $Ax = b$ and eigenvalue problems. Many iterative methods, such as the Conjugate Gradient method for symmetric positive-definite systems and the Arnoldi iteration for general eigenvalue problems, construct approximations of the solution or eigenvalues within these Krylov subspaces. The dimensions of Krylov subspaces are at most k. In practice, the vectors $v, Av, A^2v, \dots$ often become linearly dependent relatively quickly, meaning the dimension of $K_k(A, v)$ can be less than k. Algorithms like the Arnoldi iteration and the Lanczos process are designed to efficiently generate an orthonormal basis for these subspaces, which is essential for constructing and solving the reduced eigenvalue problems or linear systems that approximate the original problem. The efficiency of Krylov subspace methods stems from their ability to implicitly work with the matrix A without requiring explicit matrix-vector products for all terms in the sequence.