RQ decomposition from QR decomposition

Q and R matrices in the QR decomposition of A and the RQ decomposition of a different matrix. He has found some MATLAB code that can convert RQ to QR, but he doesn't understand how the operations are done in reverse. He also asks for tips on understanding the relationship between these factorizations. In summary, Dave is trying to understand the intuition behind the Q and R matrices in both QR and RQ decompositions and is looking for tips to help him understand the relationship between these factorizations.
  • #1
daviddoria
97
0
What I'm wondering is:

Q and R in the QR decomposition of A are the same Q and R in the RQ decomposition of which matrix?

I found some MATLAB code which will get RQ from QR, but I don't understand how you would do those operations FIRST, then find the QR decomposition.

Code: 
ReverseRows = [0 0 1; 0 1 0 ; 1 0 0];
[Q R] = qr((ReverseRows * A)');
R = ReverseRows * R' * ReverseRows;
Q = ReverseRows * Q';

Any tips about the intuition of the relations of these factorizations?

Thanks,
Dave
 
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  • #2
I guess what I was looking for is that QR is Gram-Schmidt applied to the column of A, where RQ is Gram-Schmidt applied to the rows of A.

Dave
 

Related to RQ decomposition from QR decomposition

1. What is RQ decomposition and how does it relate to QR decomposition?

RQ decomposition is a matrix decomposition technique that decomposes a matrix A into an upper triangular matrix R and an orthogonal matrix Q. This is similar to QR decomposition, where a matrix A is decomposed into an orthogonal matrix Q and an upper triangular matrix R. The difference between the two is the order in which the decomposition is performed - in RQ decomposition, the R matrix is upper triangular, while in QR decomposition, the R matrix is lower triangular.

2. What are the advantages of using RQ decomposition over QR decomposition?

RQ decomposition is advantageous in certain cases because it can be computed more efficiently than QR decomposition. This is because the multiplication of an orthogonal matrix with a triangular matrix is simpler and requires fewer operations than the multiplication of two triangular matrices. Additionally, RQ decomposition can be used for certain types of matrices, such as upper triangular matrices, which cannot be decomposed using QR decomposition.

3. How is RQ decomposition used in practical applications?

RQ decomposition is commonly used in statistics and data analysis, specifically in regression analysis. It can also be used for solving systems of linear equations, eigenvalue problems, and matrix inversion. In computer graphics, RQ decomposition is used for 3D pose estimation and camera calibration. It is also a key component in algorithms for matrix completion and low-rank matrix recovery.

4. Can RQ decomposition be applied to non-square matrices?

Yes, RQ decomposition can be applied to non-square matrices. However, the R matrix will have the same number of rows as the input matrix A, while the Q matrix will have the same number of columns. This means that for a non-square matrix, the R matrix will not be a square matrix, and the Q matrix will not be an orthogonal matrix.

5. Are there any limitations or drawbacks of using RQ decomposition?

One limitation of RQ decomposition is that it can only be applied to full rank matrices. This means that if the input matrix A is not full rank, the decomposition cannot be performed. Additionally, RQ decomposition is not unique - there can be multiple combinations of R and Q matrices that can result in the same input matrix A. This can make the interpretation and application of RQ decomposition results more challenging.

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