---
tag: FKPPWF87TR
kind: definition
name: Krylov subspace
msc: [65F10, 65F15, 15A03]
defines: QB45PCGQPV
uses: [0VD9ZSWWJG, 0DTY8B9HMS, 00YC1D652K, 03NDVCSHSB, 0EB9N0S33C]
sources:
  - url: https://en.wikipedia.org/wiki/Krylov_subspace
---

## Definition

Let $F$ be $\R$ or $\C$, let $A$ be an $n \times n$ [[matrix|0VD9ZSWWJG]] over $F$, let $b \in F^n$, and let $r \ge 1$ be an integer.
The *Krylov subspace* of order $r$ generated by $A$ and $b$ is the [[linear subspace|00YC1D652K]]
$$\mathcal{K}_r(A, b) = \operatorname{span} \set{b, A b, A^2 b, \dots, A^{r-1} b} \subseteq F^n.$$

## Remarks

Equivalently, $\mathcal{K}_r(A, b) = \set{p(A) b : p \in F[t], \ \deg p < r}$.
The subspaces grow, $\mathcal{K}_1 \subseteq \mathcal{K}_2 \subseteq \cdots$, and their dimensions increase by one until they stop growing.
Once $\mathcal{K}_{r+1} = \mathcal{K}_r$, the subspace is invariant under $A$ and all later ones are equal to it.
This happens at the least degree $d$ of a monic polynomial $p$ with $p(A) b = 0$.
That degree is at most the degree of the [[minimal polynomial|03NDVCSHSB]] of $A$, hence at most $n$.
If $A$ is [[invertible|0EB9N0S33C]], then $A^{-1} b$ lies in $\mathcal{K}_n(A, b)$, since the minimal polynomial of $A$ then has nonzero constant term.
Example: for $A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $b = e_2$, one has $A b = e_1$, so $\mathcal{K}_2(A, b) = F^2$.
Non-example: for $A = I$, every Krylov subspace $\mathcal{K}_r(I, b)$ is the line through $b$ or $\set{0}$.
So for $n \ge 2$ the whole space $F^n$ is not a Krylov subspace of the identity.