{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1SYY57DCWT"
    },
    {
      "field": "uses",
      "target": "24YCKE2RPH"
    },
    {
      "field": "uses",
      "target": "3HRHSHTG9C"
    },
    {
      "field": "uses",
      "target": "7RRB4PCRJB"
    },
    {
      "field": "uses",
      "target": "A3FR6PNEPB"
    },
    {
      "field": "uses",
      "target": "AFXZGQB4EN"
    },
    {
      "field": "uses",
      "target": "CJ5EC820S7"
    },
    {
      "field": "uses",
      "target": "HGF11AHX5R"
    },
    {
      "field": "uses",
      "target": "MA64H1RTP1"
    },
    {
      "field": "uses",
      "target": "PE5H892XVS"
    },
    {
      "field": "about",
      "target": "3HRHSHTG9C"
    },
    {
      "field": "about",
      "target": "7RRB4PCRJB"
    },
    {
      "field": "about",
      "target": "A3FR6PNEPB"
    },
    {
      "field": "about",
      "target": "AFXZGQB4EN"
    },
    {
      "field": "about",
      "target": "CJ5EC820S7"
    },
    {
      "field": "defines",
      "target": "FKPPWF87TR"
    }
  ],
  "body": "\n## Summary\n\nThe Krylov subspace of order $r$ generated by a square matrix $A$ and a vector $b$ is the [[linear span|0DTY8B9HMS]] of $b, A b, \\dots, A^{r-1} b$.\nIts elements are the vectors $p(A) b$ for polynomials $p$ of degree less than $r$.\nMany iterative methods for large linear systems and eigenvalue problems search for approximations in Krylov subspaces.\nThese subspaces can be built with matrix-vector products alone.\n\n## Explanations\n\n### L1 human\n\nStart with a vector $b$ and keep applying the matrix: $b$, $A b$, $A^2 b$, and so on.\nThe Krylov subspace is everything one can build from the first few of these vectors by adding and scaling.\nA method that can only multiply by $A$ sees exactly these vectors, so it is natural to search for the answer there.\nFor 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 $\\R^2$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "65F10",
      "65F15",
      "15A03"
    ],
    "name": "Krylov subspace",
    "same_as": {
      "wikidata": "Q1757151",
      "wikipedia": "Krylov subspace"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Krylov_subspace"
      }
    ],
    "specializes": [
      "00YC1D652K"
    ],
    "tag": "QB45PCGQPV"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/QB45PCGQPV/",
  "links": [
    {
      "field": "specializes",
      "target": "00YC1D652K"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QB45PCGQPV.md",
  "msc": [
    "65F10",
    "65F15",
    "15A03"
  ],
  "name": "Krylov subspace",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QB4/QB45PCGQPV.md",
  "status": null,
  "tag": "QB45PCGQPV",
  "uri": "m2://QB45PCGQPV",
  "url": "https://tamnd.github.io/math2/wiki/krylov-subspace/",
  "writing": {
    "tag": "QB45PCGQPV",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}