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In a coursebook I'm reading, some algorithms used to calculate the QR-decomposition of a matrix are introduced, namely Gram-Schmidt orthogonalization and the use of Givens rotations with or without column pivoting.

A question that has been posed on previous exams asks how the QR-decomposition can be used to do low-rank approximation of a matrix. This is unclear to me. I understand how this can be done using singular value decomposition, but cannot seem to grasp how the same can be achieved using the QR-decomposition.

Thanks in advance!

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    $\begingroup$ I'd be curious to see the exact wording of the question. $\endgroup$
    – littleO
    Dec 26, 2019 at 1:25
  • $\begingroup$ @littleO the exact wording of the question as it appeared on the previous exam is unknown to me. All I have is the following recollection of the question by a student that has done the exam in the past: "Discuss (including the algorithm) the QR-decomposition of a matrix using column pivoting. How can this be used to derive a low-rank approximation? Do you know any other methods to achieve a good low-rank approximation?" $\endgroup$
    – Akela Videos
    Dec 26, 2019 at 18:49
  • $\begingroup$ hmm, I'm not sure what to make of that question. Using the QR factorization of $A$ to obtain a low rank approximation of $A$ is an unusual thing to do. But you should certainly understand how to use the SVD of $A$ to obtain a low rank approximation of $A$. $\endgroup$
    – littleO
    Dec 26, 2019 at 23:44
  • $\begingroup$ @littleO I feel the same way. Using the SVD is something I can understand, as there is a lot of supporting theory that I can get behind. For the QR decomposition, it is still unclear to me how it can be used to do low-rank approximation. Anyway, thank you for your interest in my question. $\endgroup$
    – Akela Videos
    Dec 30, 2019 at 11:11

1 Answer 1

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You want a "rank revealing" QR decomposition, see https://math.berkeley.edu/~mgu/MA273/Strong_RRQR.pdf or https://people.csail.mit.edu/yujia/assets/pdf/rrqr_slides.pdf or https://www.irisa.fr/sage/wg-statlin/WORKSHOPS/LEMASSOL05/SLIDES/QR/Guyomarch.pdf (or even how rank-revealing QR factorization determine the rank of the matrix)

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    $\begingroup$ These are not bad links, but a good answer should be able to stand on its own. You could summarize and state the central equations and principles. $\endgroup$
    – Carl Christian
    Dec 26, 2019 at 1:26

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