# Effect of orthonormal projection on rank

1. Sep 14, 2009

### kalleC

1. The problem statement, all variables and given/known data
Given rank(R) and a QR factorization A = QR, what is the rank(A)

2. Relevant equations

3. 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.

2. Sep 14, 2009

### Dick

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. Sep 14, 2009

### kalleC

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. Sep 14, 2009

### Dick

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. Sep 14, 2009

### kalleC

Thank you very much =)