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So if I have an $m\times n$ matrix $A$ and I represent that matrix as $\displaystyle A = QR$, how do I write $A^{T}$ (transpose) in terms of the original $\displaystyle QR$? Does it become $\displaystyle Q^{T}R^T$ or $\displaystyle R^{T}Q^{T}$? or something else? What about for the inverse?

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$(AB)^T=B^TA^T$, provided $AB$ is definable and we are allowed to multiply $B^T$ by $A^T$ –  Babak S. Feb 21 '13 at 12:34

1 Answer 1

$(AB)^T=B^TA^T$.

That's because

$$A^T_{ij}=A_{ji}=\sum_{k=1}^{m}Q_{jk}R_{ki}=\sum_{k=1}^m R_{ik}^TQ_{kj}^T=(R^TQ^T)_{ij}$$

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