---
tag: QB45PCGQPV
kind: concept
name: Krylov subspace
msc: [65F10, 65F15, 15A03]
specializes: [00YC1D652K]
sources:
  - url: https://en.wikipedia.org/wiki/Krylov_subspace
same_as:
  wikidata: Q1757151
  wikipedia: Krylov subspace
---

## Summary

The Krylov subspace of order $r$ generated by a square matrix $A$ and a vector $b$ is the [[linear span|0DTY8B9HMS]] of $b, A b, \dots, A^{r-1} b$.
Its elements are the vectors $p(A) b$ for polynomials $p$ of degree less than $r$.
Many iterative methods for large linear systems and eigenvalue problems search for approximations in Krylov subspaces.
These subspaces can be built with matrix-vector products alone.

## Explanations

### L1 human

Start with a vector $b$ and keep applying the matrix: $b$, $A b$, $A^2 b$, and so on.
The Krylov subspace is everything one can build from the first few of these vectors by adding and scaling.
A method that can only multiply by $A$ sees exactly these vectors, so it is natural to search for the answer there.
For example, if $A$ is diagonal with entries $1$ and $2$ and $b = (1, 1)$, then $A b = (1, 2)$, and the subspace of order $2$ is all of $\R^2$.