{
  "aliases": [
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  "body": "\n## Statement\n\nLet $F$ be a field and let $A$ be an $n \\times n$ matrix over $F$.\nThen there are unique monic nonconstant polynomials $f_1, \\dots, f_r \\in F[t]$ with $f_1 \\mid f_2 \\mid \\cdots \\mid f_r$ and the following property. The matrix $A$ is [[similar|0E89HQATM8]] to the [[block diagonal matrix|001T5FJDC1]]\n$$\\begin{pmatrix} C(f_1) \u0026 \u0026 \\\\ \u0026 \\ddots \u0026 \\\\ \u0026 \u0026 C(f_r) \\end{pmatrix},$$\nwhere $C(f_i)$ is the [[companion matrix|0ZCWPNZTQ3]] of $f_i$.\nThis block matrix is the *rational canonical form* of $A$.\n\n## Remarks\n\nThe polynomials $f_1, \\dots, f_r$ are the *invariant factors* of $A$.\nThe last factor $f_r$ is the minimal polynomial of $A$, and $f_1 f_2 \\cdots f_r$ is its characteristic polynomial.\nUnlike the Jordan form, it needs no field extension: it works over any field.\nTwo matrices over $F$ are similar if and only if they have the same invariant factors.\n\n## Explanations\n\n### L1 human\n\nThe rational canonical form is a standard representative of each similarity class, over any field.\nIt uses companion matrices instead of Jordan blocks.\nIt needs no roots of polynomials, only divisibility of polynomials.\n",
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    "name": "Rational canonical form",
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      "wikidata": "Q1469423",
      "wikipedia": "Frobenius normal form"
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        "locator": "General case and theory",
        "url": "https://en.wikipedia.org/wiki/Frobenius_normal_form"
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  "markdown": "https://math2.tamnd.com/node/042JWAR3Q0.md",
  "msc": [
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  "name": "Rational canonical form",
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