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  "body": "\n## Statement\n\nLet $A$ be an $n \\times n$ complex matrix.\nThen there are a [[unitary matrix|0AR68JTMDT]] $U$ and a positive semidefinite Hermitian matrix $P$ with\n$$A = UP.$$\nThe factor $P$ is unique: $P = (A^* A)^{1/2}$, the [[square root|0BJVDCF38Y]] of $A^* A$.\nIf $A$ is invertible, then $U$ is also unique.\n\n## Remarks\n\nIt follows from the singular value decomposition $A = W \\Sigma V^*$: take $U = W V^*$ and $P = V \\Sigma V^*$.\nFor $n = 1$, it is the polar form $z = e^{i\\theta} r$ of a complex number.\nThere is also a left polar decomposition $A = P' U$ with $P' = (A A^*)^{1/2}$.\n\n## Explanations\n\n### L1 human\n\nEvery square matrix is a stretch along perpendicular axes, followed by a rotation, possibly with a reflection.\nThis generalises writing a complex number as a positive length times a unit complex number.\nThe stretch part is always unique.\n",
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