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  "body": "\n## Statement\n\nLet $A$ be an $n \\times n$ complex matrix.\nThen there are a [[unitary matrix|0AR68JTMDT]] $Q$ and an [[upper triangular matrix|0N4XY7Z3NN]] $R$ with\n$$A = QR.$$\nIf $A$ is real, then $Q$ and $R$ can be chosen real, with $Q$ orthogonal.\nIf $A$ is invertible and the diagonal entries of $R$ are required to be positive, then $Q$ and $R$ are unique.\n\n## Remarks\n\nFor an invertible $A$, the [[Gram–Schmidt process|0VT6P0GNQR]] applied to the columns of $A$ gives the columns of $Q$.\nA rectangular $m \\times n$ matrix with $m \\ge n$ has a decomposition $A = QR$ with $Q$ unitary $m \\times m$ and $R$ upper triangular $m \\times n$.\nHouseholder reflections and Givens rotations compute it stably.\nIt is used to solve least squares problems and in the QR algorithm for eigenvalues.\n\n## Explanations\n\n### L1 human\n\nThe QR decomposition writes a matrix as a rotation, possibly with reflections, times a triangular matrix.\nThe columns of $Q$ are orthonormal, and $R$ records how to rebuild the columns of $A$ from them.\nIt is the matrix form of the Gram–Schmidt process.\n",
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