{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "62NS7PV1M9"
    },
    {
      "field": "uses",
      "target": "62NS7PV1M9"
    },
    {
      "field": "defines",
      "target": "ZEZPYJC06B"
    }
  ],
  "body": "\n## Summary\n\nA square matrix is irreducible when no simultaneous permutation of its rows and columns puts it into block upper triangular form.\nFor a nonnegative matrix this means that its directed graph is strongly connected: every index can reach every other index.\nIrreducibility is the main hypothesis of the [[Perron–Frobenius theorem|62NS7PV1M9]] for nonnegative matrices that are not positive.\n\n## Explanations\n\n### L1 human\n\nRead an $n \\times n$ matrix as a network: there is an arrow from $i$ to $j$ when the entry $a_{ij}$ is not zero.\nThe matrix is irreducible when you can travel along arrows from any node to any other node.\nThen the matrix does not split into a part that never feeds back into the rest.\nFor example, $\\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ has arrows $1 \\to 2$ and $2 \\to 1$, so it is irreducible.\nThe matrix $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ has no arrow from $2$ to $1$, so it is reducible.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "15B48",
      "05C50"
    ],
    "name": "Irreducible matrix",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem"
      }
    ],
    "specializes": [
      "0JXZKX526R"
    ],
    "tag": "7Q60BB01YG"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/7Q60BB01YG/",
  "links": [
    {
      "field": "specializes",
      "target": "0JXZKX526R"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7Q60BB01YG.md",
  "msc": [
    "15B48",
    "05C50"
  ],
  "name": "Irreducible matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7Q6/7Q60BB01YG.md",
  "status": null,
  "tag": "7Q60BB01YG",
  "uri": "m2://7Q60BB01YG",
  "url": "https://math2.tamnd.com/wiki/irreducible-matrix/",
  "writing": {
    "tag": "7Q60BB01YG",
    "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
      }
    ]
  }
}