[R] chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails

[Peter Dalgaard]

Charles C. Berry wrote:
> On Sat, 21 Nov 2009, simona.racioppi at libero.it wrote:
> 
>> Hi Everyone,
>>
>> I need to take the square root of the following matrix:
>>
>>                [,1]               [,2]                [,3]
>> [1,]  0.5401984 -0.3998675 -1.3785897
>> [2,] -0.3998675  1.0561872  0.8158639
>> [3,] -1.3785897  0.8158639  1.6073119
>>
>> I tried Choleski which fails. I then tried Choleski with pivoting, but
>> unfortunately the square root I get is not valid. I also tried eigen
>> decomposition but i did no get far.
>>
>> Any clue on how to do it?!
> 
> 
> If you want to take the square root of a negative definite matrix, you 
> could use
> 
>     sqrtm( neg.def.mat )
> 
> from the expm package on rforge:
> 
>     http://r-forge.r-project.org/projects/expm/

But that matrix is not negative definite! It has 2 positive and one 
negative eigenvalue. It is non-positive definite.

It is fairly easy in any case to get a matrix square root from the eigen 
decomposition:

 > v%*%diag(sqrt(d+0i))%*%t(v)
                       [,1]                  [,2]                  [,3]
[1,]  0.5164499+0.4152591i -0.1247682-0.0562317i -0.7257079+0.3051868i
[2,] -0.1247682-0.0562317i  0.9618445+0.0076145i  0.3469916-0.0413264i
[3,] -0.7257079+0.3051868i  0.3469916-0.0413264i  1.0513849+0.2242912i
 > ch <- v%*%diag(sqrt(d+0i))%*%t(v)
 > t(ch)%*% ch
               [,1]          [,2]          [,3]
[1,]  0.5401984+0i -0.3998675-0i -1.3785897-0i
[2,] -0.3998675-0i  1.0561872+0i  0.8158639-0i
[3,] -1.3785897-0i  0.8158639-0i  1.6073119-0i

A triangular square root is, er, more difficult, but hardly impossible.

-- 
    O__  ---- Peter Dalgaard             Øster Farimagsgade 5, Entr.B
   c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
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