R中的QR分解和Choleski分解 [英] QR decomposition and Choleski decomposition in R

问题描述

我最近阅读了有关如何使用Choleski分解来计算QR分解的R矩阵的信息.关系是:

I recently read about how the R matrix of QR decomposition can be calculated using the Choleski decomposition. The relation is:

R =霍尔斯基分解(A ^ TA)

R = Choleski-decomposition(A^TA)

示例:

> A=matrix(c(1,2,3,2,3,5,1,3,2), nrow=3)
> A
     [,1] [,2] [,3]
[1,]    1    2    1
[2,]    2    3    3
[3,]    3    5    2

> AtA = t(A)%*%A
> AtA
     [,1] [,2] [,3]
[1,]   14   23   13
[2,]   23   38   21
[3,]   13   21   14

现在计算QR和Choleski分解:

Now calculating QR and Choleski decomposition:

> chol(AtA)
         [,1]     [,2]       [,3]
[1,] 3.741657 6.147009  3.4743961
[2,] 0.000000 0.462910 -0.7715167
[3,] 0.000000 0.000000  1.1547005

> qr_A = qr(A)
> qr.R(qr_A)
          [,1]      [,2]       [,3]
[1,] -3.741657 -6.147009 -3.4743961
[2,]  0.000000  0.462910 -0.7715167
[3,]  0.000000  0.000000 -1.1547005

如所观察到的，由Choleski和QR分解计算的R矩阵的值不相同. chol(AtA)的第一和第三行与qr.R(qr_A)取反.这是为什么?我假设的关系不正确吗?

As observed, the values of the R matrix calculated from Choleski and QR decomposition are not the same. The first and the third rows of chol(AtA) are negated w.r.t qr.R(qr_A). Why is that? Is the relation I'm assuming not correct?

推荐答案

矩阵的QR分解不是唯一的！有一个QR分解，其中R = chol(AtA)，但也有其他分解，qr不必给出那个.在您的示例中

The QR-decomposition of a matrix is not unique! There is a QR-decomposition with R=chol(AtA), but there are also others and qr does not necessairily give that one. In your example

qr.Q(qr_A)%*%qr.R(qr_A) 

和

(qr.Q(qr_A)%*%diag(c(-1,1,-1)))%*%chol(AtA)

都是A的有效QR分解.

are both valid QR-decompositions of A.

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