简化行梯形表格(rref)

问题描述

我发现Matlab中的rref通过空间旋转实现了高斯消除，例如，以A为5x9矩阵:

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)给出:

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

如何通过执行A的LU分解来获得与rref相同的结果?我发现LU分解也确实通过部分枢轴消除了高斯.如何用LU替换代码rref?在[L，U] = lu(A)之后获得与上述示例的rref(A)相同的结果将要执行哪些步骤.

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.

推荐答案

如果假设 A (大小为 nxm )是一个复合矩阵，例如 A =(A1 | A2)，其中 A1 是 nxn 矩阵，而 A2 是 nx(mn)矩阵.

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.

如果 A1 是可逆的，则可以编写 R ，即矩阵 A 简化行梯形形式 >:

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)

因此，您可以使用LU分解或直接的多RHS求解器来计算 A1 ^ -1 A2 .

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

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