{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0363JMQ42Y"
    },
    {
      "field": "uses",
      "target": "042JWAR3Q0"
    },
    {
      "field": "uses",
      "target": "085HQ5G5TE"
    },
    {
      "field": "uses",
      "target": "0TYMASG33Y"
    },
    {
      "field": "uses",
      "target": "0Y17PEKF24"
    }
  ],
  "body": "\n## Definition\n\nLet $F$ be a field and let $A$ and $B$ be $n \\times n$ matrices over $F$.\nThe matrix $B$ is *similar* to $A$ when there is an [[invertible|0EB9N0S33C]] $n \\times n$ matrix $P$ over $F$ with\n$$B = P^{-1} A P.$$\n\n## Remarks\n\nSimilarity is an equivalence relation on $M_n(F)$.\nBy the [[change of basis formula|0MQXVZEBMT]], two matrices are similar if and only if they represent the same endomorphism of $F^n$ in two bases.\nSimilar matrices have the same [[trace|0XC1RDBY7A]], [[determinant|00KDZRW4C0]], [[characteristic polynomial|0VSSPAFM2C]], [[rank|0ZC9YTXZRY]] and [[eigenvalues|0BQAA2BK3Q]].\nThe matrices $I_2$ and $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ have the same characteristic polynomial but are not similar.\n\n## Explanations\n\n### L1 human\n\nSimilar matrices describe the same linear map, seen from two different bases.\nProperties of the map itself, like trace and determinant, are the same for both matrices.\nA main goal of linear algebra is to find a simple matrix in each similarity class.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0E89HQATM8",
    "kind": "definition",
    "msc": [
      "15A21"
    ],
    "name": "Similar matrices",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Matrix_similarity"
      }
    ],
    "tag": "027KDW2K72",
    "uses": [
      "0TWTS88TWN"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/027KDW2K72/",
  "links": [
    {
      "field": "defines",
      "target": "0E89HQATM8"
    },
    {
      "field": "uses",
      "target": "0TWTS88TWN"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/027KDW2K72.md",
  "msc": [
    "15A21"
  ],
  "name": "Similar matrices",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/027/027KDW2K72.md",
  "status": null,
  "tag": "027KDW2K72",
  "uri": "m2://027KDW2K72",
  "url": "https://math2.tamnd.com/wiki/similar-matrices-definition/",
  "writing": {
    "tag": "027KDW2K72",
    "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
      }
    ]
  }
}