Effect of orthonormal projection on rank

  • Thread starter kalleC
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  • #1
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Homework Statement


Given rank(R) and a QR factorization A = QR, what is the rank(A)


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The Attempt at a Solution


I want to know if multiplication by a full rank orthonormal matrix Q and an upper trapezoidal matrix R yields rank(R)=rank(Q*R)=rank(A)

This is mostly guesswork by me but I'd like to use it for a question I need to answer.
 

Answers and Replies

  • #2
Dick
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Well, the rank of a matrix is the dimension of the image, right? If the image of R is a subspace S of dimension rank(R), then what's the dimension of Q(S) if Q is full rank?
 
  • #3
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They are equal? As the only way Q(S) would be dissimilar would be if rank(Q)<rank(R).

But does not the reason for this have anything to do with Q being orthnormal? Otherwise couldn't Q act on R and cause some of the image to overlap effectively reducing the rank?
 
  • #4
Dick
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Q is full rank, so it's one to one. So yes, rank(QR)=dim(Q(S))=dim(S)=rank(R). So rank(QR)=rank(R).
 
  • #5
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Thank you very much =)
 

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