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Is it possible to know whether two matrices have the same Q in the QR-decomposition without explicitly calculating the QR decomposition.

ie: $$A = Q_1R_1$$ and $$B = Q_2R_2$$

Is it possible to recognize that $$Q_1=Q_2$$ or $$Q_1 \ne Q_2$$ just by looking at A and B or their properties? (ie: is there some shortcut method to recognize this, without actually calculating the Q matrices).

Ameet Sharma
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    In principle, yes: $Q_1=Q_2$ (up to signs of the columns) if and only if $\operatorname{span}{a_1,\dots,a_i}=\operatorname{span}{b_1,\dots,b_i}$ for all $i=1,\dots,n$, where $a_i,b_i$ are the columns of $A,B$ respectively. However this may be just as hard to check as computing the QR decompositions. –  Oct 03 '19 at 08:45

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