Effect of orthonormal projection on rank

[kalleC]

Homework Statement

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

Homework Equations

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.

 

[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?

 

[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?

 

[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).

 

[kalleC]

Thank you very much =)