# If a mxn matrix A, m>=n has a reduced QR-decomposition

1. Sep 15, 2010

### Vikt0r

Hi
I am arguing with a friend about the following:
He claims that if a mxn matrix A, m>=n has a reduced QR-decomposition where R has j nonzero diagonal elements, then the rank of A is at least j. I claim that it is exactly j.
It was some years ago since i read linear algebra so i was hoping someone here could help us out.

2. Sep 19, 2010

Re: QR-decomposition

Do you really mean what you wrote? Or you mean "exactly j nonzero diagonal elements?" Because, being precise, your condition can be also understood as "where R has at least j nonzero elements, perhaps more".

3. Sep 23, 2010

### diggy

Re: QR-decomposition

Yes, A's rank should be exactly j. In a full decomposition the remaining columns in Q associated with the zero-valued R's should span A's null space. Combined they (all of Q) covers the full m space, which of course makes sense since Q is an orthonormal basis.

4. Sep 23, 2010

### diggy

Re: QR-decomposition

Another way to put it is that: rank(A) = rank(QR) = rank(R), since Q covers the full basis. And the rank of R is j.

5. Sep 23, 2010