Krylov subspace
In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A, that is,
Background
The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about the concept in 1931.Properties
- .
- Let. Then are linearly independent unless, for all, and. So is the maximal dimension of the Krylov subspaces.
- The maximal dimension satisfies and.
- Consider, where is the minimal polynomial of. We have. Moreover, for any, there exists a for which this bound is tight, i.e..
- is a cyclic submodule generated by of the torsion -module, where is the linear space on.
- can be decomposed as the direct sum of Krylov subspaces.
Use
Modern iterative methods such as Arnoldi iteration can be used for finding one eigenvalues of large sparse matrices or solving large systems of linear equations. They try to avoid matrix-matrix operations, but rather multiply vectors by the matrix and work with the resulting vectors. Starting with a vector, one computes, then one multiplies that vector by to find and so on. All algorithms that work this way are referred to as Krylov subspace methods; they are among the most successful methods currently available in numerical linear algebra. These methods can be used in situations where there is an algorithm to compute the matrix-vector multiplication without there being an explicit representation of, giving rise to matrix-free methods.