{
  "aliases": [
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  "body": "\n## Statement\n\nLet $F$ be an algebraically closed field, for example $\\C$, and let $A$ be an $n \\times n$ matrix over $F$.\nThen $A$ is [[similar|0E89HQATM8]] to a [[block diagonal matrix|001T5FJDC1]]\n$$J = \\begin{pmatrix} J_{k_1}(\\lambda_1) \u0026 \u0026 \\\\ \u0026 \\ddots \u0026 \\\\ \u0026 \u0026 J_{k_r}(\\lambda_r) \\end{pmatrix},$$\nwhose diagonal blocks are [[Jordan blocks|07PH4RX0X5]].\nThe matrix $J$ is unique up to the order of the blocks.\nIt is the *Jordan normal form* of $A$.\n\n## Remarks\n\nThe $\\lambda_i$ are the eigenvalues of $A$; they need not be distinct.\nThe number of blocks for $\\lambda$ is its geometric multiplicity, and the sum of their sizes is its algebraic multiplicity.\nThe largest block for $\\lambda$ has size equal to the multiplicity of $\\lambda$ in the minimal polynomial.\nThe theorem holds over any field $F$ for the matrices whose characteristic polynomial splits over $F$.\n\n## Explanations\n\n### L1 human\n\nEvery complex square matrix becomes almost diagonal in a good basis.\nThe only entries off the diagonal are some $1$s just above it, inside the Jordan blocks.\nTwo complex matrices are similar exactly when they have the same Jordan form, up to the order of the blocks.\n",
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    "name": "Jordan normal form",
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  "name": "Jordan normal form",
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