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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]] and let $m, n \\ge 1$ be integers.\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and give the space $\\mathbb{K}^{m \\times n}$ of $m \\times n$ [[matrices|0VD9ZSWWJG]] the Borel $\\sigma$-algebra of $\\mathbb{K}^{mn}$.\nA *random $m \\times n$ matrix* is a measurable map $X \\colon \\Omega \\to \\mathbb{K}^{m \\times n}$.\n\n## Remarks\n\nEquivalently, $X = (X_{ij})$ where each entry $X_{ij}$ is a [[random variable|0PNRKMRC55]] on the same space $(\\Omega, \\mathcal{F}, P)$.\nThe law of $X$ is the probability measure $P \\circ X^{-1}$ on $\\mathbb{K}^{m \\times n}$.\nExample: let $G$ be an $n \\times n$ matrix whose entries are independent standard normal random variables, and let $H = (G + G^T)/\\sqrt{2}$.\nEach entry of $H$ is a sum of measurable functions, so $H$ is a random symmetric matrix; it is a Wigner matrix with Gaussian entries.\nNon-example: $n^2$ random variables defined on different probability spaces do not form a random matrix, because the entries need one common probability space.\n",
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