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I am trying to understand if the ordering of columns matters in QR decompsoition.

In general it seems that column ordering won't matter. I guess for SVD or any matrix factorization the way columns and rows are ordered has no effect, i.e. we can jumble up the columns and rows entirely in linear algebra and it wont matter to algorithms.

Am I corrrect? Do orderings matter? Any relevant literature?

Cheers!

mourinho
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  • "matter" in regards to what? Runtime? Obviously if you rearrange the columns the *result* of the decompositions will change too... – jbowman Oct 10 '18 at 19:47
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    The relevant chapters of Golub & Van Loan will have the answer. TL;DR the term of art is "pivoting." – Sycorax Oct 10 '18 at 19:52
  • @jbowman no, in terms of the final results, in terms of the intermediate results as well, and also runtime. – mourinho Oct 10 '18 at 19:52
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    Just to clarify a point that probably doesn't need clarification - it will matter in terms of results, e.g., the QR decomposition of the 2x2 identity matrix is different to the QR decomposition of the 2x2 identity matrix with the columns swapped. However, in the case of both SVD and QR, the algorithms themselves engage in a fixed sequence of operations regardless of the numbers they are operating on, so in those terms it doesn't matter, unless perhaps some implementations actually check numerical values to identify special cases. – jbowman Oct 10 '18 at 20:52

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