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 (starting from ${\displaystyle A^{0}=I}$), that is,^[1][2]

${\displaystyle {\mathcal {K}}_{r}(A,b)=\operatorname {span} \,\{b,Ab,A^{2}b,\ldots ,A^{r-1}b\}.}$

The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about the concept in 1931.^[3]

• ${\displaystyle {\mathcal {K}}_{r}(A,b),A\,{\mathcal {K}}_{r}(A,b)\subset {\mathcal {K}}_{r+1}(A,b)}$.
• Let ${\displaystyle r_{0}=\operatorname {dim} \operatorname {span} \,\{b,Ab,A^{2}b,\ldots \}}$. Then ${\displaystyle \{b,Ab,A^{2}b,\ldots ,A^{r-1}b\}}$ are linearly independent unless ${\displaystyle r>r_{0}}$, ${\displaystyle {\mathcal {K}}_{r}(A,b)\subset {\mathcal {K}}_{r_{0}}(A,b)}$ for all ${\displaystyle r}$, and ${\displaystyle \operatorname {dim} {\mathcal {K}}_{r_{0}}(A,b)=r_{0}}$. So ${\displaystyle r_{0}}$ is the maximal dimension of the Krylov subspaces ${\displaystyle {\mathcal {K}}_{r}(A,b)}$.
• The maximal dimension satisfies ${\displaystyle r_{0}\leq 1+\operatorname {rank} A}$ and ${\displaystyle r_{0}\leq n}$.
• Consider ${\displaystyle \dim \operatorname {span} \,\{I,A,A^{2},\ldots \}=\deg \,p(A)}$, where ${\displaystyle p(A)}$ is the minimal polynomial of ${\displaystyle A}$. We have ${\displaystyle r_{0}\leq \deg \,p(A)}$. Moreover, for any ${\displaystyle A}$, there exists a ${\displaystyle b}$ for which this bound is tight, i.e. ${\displaystyle r_{0}=\deg \,p(A)}$.
• ${\displaystyle {\mathcal {K}}_{r}(A,b)}$ is a cyclic submodule generated by ${\displaystyle b}$ of the torsion ${\displaystyle k[x]}$-module ${\displaystyle (k^{n})^{A}}$, where ${\displaystyle k^{n}}$ is the linear space on ${\displaystyle k}$.
• ${\displaystyle k^{n}}$ can be decomposed as the direct sum of Krylov subspaces.^{[clarification needed]}

Krylov subspaces are used in algorithms for finding approximate solutions to high-dimensional linear algebra problems.^[2] 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.^[4]

Modern iterative methods such as Arnoldi iteration can be used for finding one (or a few) 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 ${\displaystyle b}$, one computes ${\displaystyle Ab}$, then one multiplies that vector by ${\displaystyle A}$ to find ${\displaystyle A^{2}b}$ 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 ${\displaystyle A}$, giving rise to Matrix-free methods.

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.

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).

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

• Nevanlinna, Olavi (1993). Convergence of iterations for linear equations. Lectures in Mathematics ETH Zürich. Basel: Birkhäuser Verlag. pp. viii+177 pp. ISBN 3-7643-2865-7. MR 1217705.
• Saad, Yousef (2003). Iterative methods for sparse linear systems (2nd ed.). SIAM. ISBN 0-89871-534-2. OCLC 51266114.
• Gerard Meurant and Jurjen Duintjer Tebbens: ”Krylov methods for nonsymmetric linear systems - From theory to computations”, Springer Series in Computational Mathematics, vol.57, (Oct. 2020). ISBN 978-3-030-55250-3, url=https://doi.org/10.1007/978-3-030-55251-0.
• Iman Farahbakhsh: "Krylov Subspace Methods with Application in Incompressible Fluid Flow Solvers", Wiley, ISBN 978-1119618683 (Sep., 2020).