{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "XEC5MF8WWX"
    }
  ],
  "body": "\n## Summary\n\nA square root of a square matrix $A$ is a matrix $B$ with $B^2 = A$.\nA matrix can have no square root, finitely many, or infinitely many.\nA [[positive semidefinite matrix|01MMVW8CKC]] has exactly one positive semidefinite square root; see [[square root of a positive semidefinite matrix|0BJVDCF38Y]].\nThis square root is used in statistics, polar decompositions and quantum mechanics.\n\n## Explanations\n\n### L1 human\n\nFor numbers, a square root of $a$ is a number $b$ with $b^2 = a$.\nFor matrices the idea is the same, with matrix multiplication.\nThe diagonal matrix with entries $4$ and $9$ has the square root with entries $2$ and $3$, and also those with entries $\\pm 2$ and $\\pm 3$.\nThe identity matrix of size $2$ even has infinitely many square roots, such as every reflection of the plane.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "kind": "concept",
    "msc": [
      "15A16",
      "15A24"
    ],
    "name": "Square root of a matrix",
    "same_as": {
      "wikidata": "Q964330",
      "wikipedia": "Square root of a matrix"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Square_root_of_a_matrix"
      }
    ],
    "tag": "63FAKMYZ2E"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/63FAKMYZ2E/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/63FAKMYZ2E.md",
  "msc": [
    "15A16",
    "15A24"
  ],
  "name": "Square root of a matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/63F/63FAKMYZ2E.md",
  "status": null,
  "tag": "63FAKMYZ2E",
  "uri": "m2://63FAKMYZ2E",
  "url": "https://math2.tamnd.com/wiki/square-root-of-a-matrix/",
  "writing": {
    "tag": "63FAKMYZ2E",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}