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

• .

• .

• .

• is the minimal polynomial.

• of the torsion.

• can be decomposed as the direct sum of Krylov subspaces.

Use

Krylov subspaces are used in algorithms for finding approximate solutions to high-dimensional linear algebra problems. Many linear dynamical system tests in control theory, especially those related to controllability and observability, involve checking the rank of the Krylov subspace. These tests are equivalent to finding the span of the Gramians associated with the system/output maps so the uncontrollable and unobservable subspaces are simply the orthogonal complement to the Krylov subspace.

Modern iterative methods such as Arnoldi iteration can be used for finding one (or a few) eigenvalues of large sparse matrices, giving rise to matrix-free methods.

Issues

Because the vectors usually soon become almost linearly dependent due to the properties of power iteration, methods relying on Krylov subspace frequently involve some orthogonalization scheme, such as Lanczos iteration for Hermitian matrices or Arnoldi iteration for more general matrices.

Existing methods

The best known Krylov subspace methods are the Conjugate gradient, IDR(s) (Induced dimension reduction), GMRES (generalized minimum residual), BiCGSTAB (biconjugate gradient stabilized), QMR (quasi minimal residual), TFQMR (transpose-free QMR) and MINRES (minimal residual method).

See also

• Iterative method, which has a section on Krylov subspace methods

References

Further reading

• Charles George Broyden and Maria Teresa Vespucci(2004): Krylov Solvers for Linear Algebraic Systems, Elsevier(Studies in Computational Mathematics 11), ISBN 0-444-51474-0.

• Iman Farahbakhsh: Krylov Subspace Methods with Application in Incompressible Fluid Flow Solvers, Wiley, (Sep., 2020).