Reduced Row Echelon Form (rref)

问题

I found that rref in Matlab does the gaussian elimination with patial pivoting Let A for example be a 5x9 matrix:

0.4898    0.2760    0.4984    0.7513    0.9593    0.8407    0.3500    0.3517    0.2858
0.4456    0.6797    0.9597    0.2551    0.5472    0.2543    0.1966    0.8308    0.7572
0.6463    0.6551    0.3404    0.5060    0.1386    0.8143    0.2511    0.5853    0.7537
0.7094    0.1626    0.5853    0.6991    0.1493    0.2435    0.6160    0.5497    0.3804
0.7547    0.1190    0.2238    0.8909    0.2575    0.9293    0.4733    0.9172    0.5678

rref(A) gives:

1.0000         0         0         0         0   10.9716   -6.2494   33.3062   16.0275
     0    1.0000         0         0         0   -2.2910    1.6003   -9.5889   -3.9001
     0         0    1.0000         0         0   -3.3952    1.8012   -6.8843   -3.4078
     0         0         0    1.0000         0   -8.3071    5.8617  -27.3981  -13.0805
     0         0         0         0    1.0000    4.2036   -2.4313   11.1545    5.2517

How to obtain the same result as rref by performing LU factorization of A? I found that LU factorization does gauss elimination with partial pivoting as well. How to replace in a code rref with LU? What are the steps to do after [L,U]=lu(A) to obtain the same result as rref(A) of the above example.

回答1:

If you assume A (of size n x m) is a compound matrix such as A = (A1 | A2) with A1 a n x n matrix and A2 a n x (m-n) matrix.

If A1 is invertible, then you can write R, the Reduced Row Echelon Form of the matrix A:

R = A1^-1 (A1 | A2) = (Id | A1^-1 A2)

You can thus use a LU decomposition or a direct multi-RHS solver to compute A1^-1 A2.

来源：https://stackoverflow.com/questions/28700469/reduced-row-echelon-form-rref