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  "body": "\n## Statement\n\nLet $A$ be an $m \\times n$ complex matrix.\nThen there are a [[unitary matrix|0AR68JTMDT]] $U$ of size $m \\times m$, a unitary matrix $V$ of size $n \\times n$, and an $m \\times n$ matrix $\\Sigma$ with\n$$A = U \\Sigma V^*.$$\nHere $\\Sigma_{ij} = 0$ for $i \\ne j$, and $\\Sigma_{11} \\ge \\Sigma_{22} \\ge \\cdots \\ge 0$ are the [[singular values|09VMDSRXWJ]] of $A$.\n\n## Remarks\n\nIf $A$ is real, then $U$ and $V$ can be chosen real orthogonal.\nThe columns of $V$ are eigenvectors of $A^* A$, and the columns of $U$ are eigenvectors of $A A^*$.\nThe matrix $\\Sigma$ is unique; $U$ and $V$ are not unique in general.\nKeeping the $k$ largest singular values gives the best approximation of $A$ of rank $k$, for the Frobenius and operator norms.\n\n## Explanations\n\n### L1 human\n\nEvery matrix is a rotation, then a stretch along the axes, then another rotation.\nThe stretch factors are the singular values.\nThis decomposition is a central tool in data analysis and numerical computation.\n",
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