Krylov subspace

In

${\displaystyle A^{0}=I}$

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

Background

The concept is named after Russian applied mathematician and naval engineer

• ${\displaystyle {\mathcal {K}}_{r}(A,b),A\,{\mathcal {K}}_{r}(A,b)\subset {\mathcal {K}}_{r+1}(A,b)}$.
• 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]}

Use

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.

Issues

Because the vectors usually soon become almost

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

References