For $A\in\mathbb C^{m\times n}$ of full column rank with $m\ge n$, a reduced QR factorization is $A=QR$, where $Q^*Q=I_n$ and $R$ is invertible upper triangular. In a full QR factorization, $Q$ is $m\times m$ unitary and the rectangular $R$ includes zero lower rows. Real matrices use orthogonal $Q$.
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