[R] about the Choleski factorization

[Duncan Murdoch]

On 3/27/2009 11:46 AM, 93354504 wrote:
> Hi there, 
> 
> Given a positive definite symmetric matrix, I can use chol(x) to obtain U where U is upper triangular
> and x=U'U. For example,
> 
> x=matrix(c(5,1,2,1,3,1,2,1,4),3,3)
> U=chol(x)
> U
> #         [,1]      [,2]      [,3]
> #[1,] 2.236068 0.4472136 0.8944272
> #[2,] 0.000000 1.6733201 0.3585686
> #[3,] 0.000000 0.0000000 1.7525492
> t(U)%*%U   # this is exactly x
> 
> Does anyone know how to obtain L such that L is lower triangular and x=L'L? Thank you.
> 
> Alex
> 
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> and provide commented, minimal, self-contained, reproducible code.

 > rev <- matrix(c(0,0,1,0,1,0,1,0,0),3,3)
 > rev
      [,1] [,2] [,3]
[1,]    0    0    1
[2,]    0    1    0
[3,]    1    0    0

(the matrix that reverses the row and column order when you pre and post 
multiply it).

Then

L <- rev %*% chol(rev %*% x %*% rev) %*% rev

is what you want, i.e. you reverse the row and column order of the 
Choleski square root of the reversed x:

 > x
      [,1] [,2] [,3]
[1,]    5    1    2
[2,]    1    3    1
[3,]    2    1    4

 > L <- rev %*% chol(rev %*% x %*% rev) %*% rev
 > L
           [,1]     [,2] [,3]
[1,] 1.9771421 0.000000    0
[2,] 0.3015113 1.658312    0
[3,] 1.0000000 0.500000    2
 > t(L) %*% L
      [,1] [,2] [,3]
[1,]    5    1    2
[2,]    1    3    1
[3,]    2    1    4

Duncan Murdoch