# [R] about the Choleski factorization

Peter Dalgaard p.dalgaard at biostat.ku.dk
Fri Mar 27 19:07:28 CET 2009

```Duncan Murdoch wrote:
> 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
>>
>> ______________________________________________
>> R-help at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide
>> http://www.R-project.org/posting-guide.html
>> 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

Or just

> r<-3:1
> chol(x[r,r])[r,r]
[,1]     [,2] [,3]
[1,] 1.9771421 0.000000    0
[2,] 0.3015113 1.658312    0
[3,] 1.0000000 0.500000    2

(It is after all, just a matter of starting from the other end).

--
O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
(*) \(*) -- University of Copenhagen   Denmark      Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)              FAX: (+45) 35327907

```