{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "01APZ8GDQH"
    },
    {
      "field": "uses",
      "target": "02KYGSCTCN"
    },
    {
      "field": "uses",
      "target": "0YB93GWTKK"
    }
  ],
  "body": "\n## Definition\n\nLet $S$ be a nonempty countable set.\nA transition matrix, or stochastic matrix, on $S$ is a family $P = (p_{ij})_{i, j \\in S}$ of real numbers with\n$$p_{ij} \\ge 0 \\quad \\text{for all } i, j \\in S \\quad \\text{and} \\quad \\sum_{j \\in S} p_{ij} = 1 \\quad \\text{for each } i \\in S.$$\nThe transition matrix of a time-homogeneous [[Markov chain|033TJBZDG0]] $(X_n)$ on $S$ has $p_{ij} = P(X_{n+1} = j \\mid X_n = i)$.\n\n## Remarks\n\nEach row of $P$ is a probability distribution on $S$: the law of the next state, given the present state $i$.\nA product of two stochastic matrices is stochastic. The $n$-th power $P^n$ gives the $n$-step probabilities $P(X_{m+n} = j \\mid X_m = i)$.\nIf the row vector $\\lambda$ is the law of $X_0$, then $\\lambda P^n$ is the law of $X_n$.\nA finite stochastic matrix has the eigenvalue $1$, with the eigenvector $(1, \\dots, 1)^{\\mathsf{T}}$, and all its eigenvalues have absolute value at most $1$.\n\n## Explanations\n\n### L1 human\n\nThe transition matrix is a table of the chances to move from each state to each state in one step.\nFor weather with states \"sun\" and \"rain\", the row of \"sun\" gives the chances of sun and of rain tomorrow, if today is sunny.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0F462493EA",
    "kind": "definition",
    "msc": [
      "60J10"
    ],
    "name": "Transition matrix",
    "sources": [
      {
        "locator": "Definition and properties",
        "url": "https://en.wikipedia.org/wiki/Stochastic_matrix"
      }
    ],
    "tag": "0F1KBVY161",
    "uses": [
      "033TJBZDG0"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0F1KBVY161/",
  "links": [
    {
      "field": "defines",
      "target": "0F462493EA"
    },
    {
      "field": "uses",
      "target": "033TJBZDG0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0F1KBVY161.md",
  "msc": [
    "60J10"
  ],
  "name": "Transition matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0F1/0F1KBVY161.md",
  "status": null,
  "tag": "0F1KBVY161",
  "uri": "m2://0F1KBVY161",
  "url": "https://math2.tamnd.com/wiki/transition-matrix-definition/",
  "writing": {
    "tag": "0F1KBVY161",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}