about the Choleski factorization

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

 > 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