R help - Nov 2009 - R: RE: R: Re: R: Re: chol( neg.def.matrix ) WAS: Re: Choleski and Choleski with pivoting of matrix fails

Thanks for your message!
Actually it works quite well for me too.

If I then take the trace of the final result below, I end up with a number 
made up of both a real and an imaginary part. This does not probably mean much 
if the trace of the matrix below givens me info about the degrees of freedom of 
a model...

Simona 
>----Messaggio originale----
>Da: RVaradhan at jhmi.edu
>Data: 25-nov-2009 18.55
>A: <simona.racioppi at libero.it>, <P.Dalgaard at biostat.ku.dk>
>Cc: <r-help at r-project.org>
>Ogg: RE: [R] R: Re: R: Re: chol( neg.def.matrix ) WAS: Re: Choleski and 
Choleski with pivoting of matrix fails
>
>I do not understand what the problem is, as it works just fine for me:
>
>A <- matrix(c(0.5401984,-0.3998675,-1.3785897,-0.3998675,1.0561872,  
>0.8158639,-1.3785897, 0.8158639, 1.6073119), 3, 3, byrow=TRUE)
>
>eA <- eigen(A)
>
>chA <-  eA$vec %*% diag(sqrt(eA$val+0i)) %*% t(eA$vec)
>
>all.equal(A, Re(chA %*% t(chA)))
>
>Y <- diag(c(1,2,3))
>
>solve(chA %*% Y)
>
>Ravi.
>
>-----------------------------------------------------------------------------------
>
>Ravi Varadhan, Ph.D.
>
>Assistant Professor, The Center on Aging and Health
>
>Division of Geriatric Medicine and Gerontology 
>
>Johns Hopkins University
>
>Ph: (410) 502-2619
>
>Fax: (410) 614-9625
>
>Email: rvaradhan at jhmi.edu
>
>Webpage:  http://www.jhsph.
edu/agingandhealth/People/Faculty_personal_pages/Varadhan.html
>
> 
>
>------------------------------------------------------------------------------------
>
>-----Original Message-----
>From: r-help-bounces at r-project.org [mailto:r-help-bounces at
r-project.org] On
Behalf Of simona.racioppi at libero.it
>Sent: Wednesday, November 25, 2009 9:59 AM
>To: P.Dalgaard at biostat.ku.dk
>Cc: r-help at r-project.org
>Subject: [R] R: Re: R: Re: chol( neg.def.matrix ) WAS: Re: Choleski and 
Choleski with pivoting of matrix fails
>
>Dear Peter,
>thank you very much for your answer.
>
>My problem is that I need to calculate the following quantity:
>
>solve(chol(A)%*%Y)
>
>Y is a 3*3 diagonal matrix and A is a 3*3 matrix. Unfortunately one 
>eigenvalue of A is negative. I can anyway take the square root of A but when
I 
>multiply it by Y, the imaginary part of the square root of A is dropped, and
I 
>do not get the right answer.
>
>I tried to exploit the diagonal structure of Y by using 2*2 matrices for A 
>and Y. In this way the problem mentioned above disappears (since all 
>eigenvalues of A are positive) and when I perform the calculation above I 
get 
>approximately the right answer. The approximation is quite good. However it 
is 
>an approximation.
>
>Any suggestion?
>Thank you very much!
>Simon
>
>
>
>
>>----Messaggio originale----
>>Da: P.Dalgaard at biostat.ku.dk
>>Data: 23-nov-2009 14.09
>>A: "simona.racioppi at libero.it"<simona.racioppi at
libero.it>
>>Cc: "Charles C. Berry"<cberry at tajo.ucsd.edu>,
<r-help at r-project.org>
>>Ogg: Re: R: Re: [R] chol( neg.def.matrix ) WAS: Re: Choleski and
Choleski
>with pivoting of matrix fails
>>
>>simona.racioppi at libero.it wrote:
>>> It works! But Once I have the square root of this matrix, how do I 
convert 
>it 
>>> to a real (not imaginary) matrix which has the same property? Is
that
>>> possible?
>>
>>No. That is theoretically impossible.
>>
>>If A = B'B, then x'Ax = ||Bx||^2 >= 0
>>
>>for any x, which implies in particular that all eigenvalues of A should
>>be nonnegative.
>>
>>> 
>>> Best,
>>> Simon
>>> 
>>>> ----Messaggio originale----
>>>> Da: p.dalgaard at biostat.ku.dk
>>>> Data: 21-nov-2009 18.56
>>>> A: "Charles C. Berry"<cberry at tajo.ucsd.edu>
>>>> Cc: "simona.racioppi at libero.it"<simona.racioppi
at libero.it>, <r-help at r-
>>> project.org>
>>>> Ogg: Re: [R] chol( neg.def.matrix ) WAS: Re: Choleski and
Choleski
with 
>>> pivoting of matrix fails
>>>> 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             ?ster 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
>>>>
>>> 
>>> 
>>
>>
>>-- 
>>   O__  ---- Peter Dalgaard             ?ster 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
>>
>>
>
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