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

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In summary, the conversation is about the rank of a mxn matrix A with m>=n and its reduced QR-decomposition where R has j nonzero diagonal elements. One person claims that the rank of A is at least j while the other claims it is exactly j. It is also mentioned that Q is an orthonormal or unitary matrix and that the rank of R is j.
  • #1
Vikt0r
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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.
 
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  • #2


Vikt0r said:
Hi
where R has j nonzero diagonal elements

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


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


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


Then, you are right. To say "Q covers the full basis" is a little bit not quite precise. Q is an orthogonal (or unitary) matrix, thus, in particular, invertible. That is all you need.
 

1. What is a reduced QR-decomposition for a mxn matrix?

A reduced QR-decomposition for a mxn matrix A, where m>=n, is a factorization of A into the product of an m×n matrix Q and an n×n upper triangular matrix R.

2. How is a reduced QR-decomposition different from a regular QR-decomposition?

A reduced QR-decomposition differs from a regular QR-decomposition in that the Q matrix in a reduced QR-decomposition has orthonormal columns, while in a regular QR-decomposition, the Q matrix has orthogonal columns.

3. What are the applications of a reduced QR-decomposition in scientific research?

A reduced QR-decomposition has many applications in scientific research, including solving linear systems of equations, computing eigenvalues and eigenvectors, and performing least squares regressions.

4. What are the benefits of using a reduced QR-decomposition over other matrix factorizations?

One of the main benefits of using a reduced QR-decomposition is that it is numerically stable and efficient for solving a variety of problems, such as linear systems and least squares problems. It also allows for easier computation of eigenvalues and eigenvectors compared to other factorizations.

5. How is a reduced QR-decomposition calculated?

A reduced QR-decomposition can be calculated using the Gram-Schmidt process, which involves orthogonalizing the columns of the original matrix A to create the Q matrix, and then using a modified version of Gaussian elimination to obtain the upper triangular matrix R.

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