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: rvarad...@jhmi.edu
Webpage:
http://www.jhsph.edu/agingandhealth/People/Faculty_personal_pages/Varadhan.html
-Original Message-
From: r-help-boun...@r-project.org [mailto:r-help-boun...@r-project.org] On
Behalf Of simona.racio...@libero.it
Sent: Wednesday, November 25, 2009 9:59 AM
To: p.dalga...@biostat.ku.dk
Cc: r-help@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.dalga...@biostat.ku.dk
Data: 23-nov-2009 14.09
A: simona.racio...@libero.itsimona.racio...@libero.it
Cc: Charles C. Berrycbe...@tajo.ucsd.edu, r-help@r-project.org
Ogg: Re: R: Re: [R] chol( neg.def.matrix ) WAS: Re: Choleski and Choleski
with pivoting of matrix fails
simona.racio...@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.dalga...@biostat.ku.dk
Data: 21-nov-2009 18.56
A: Charles C. Berrycbe...@tajo.ucsd.edu
Cc: simona.racio...@libero.itsimona.racio...@libero.it, r-h...@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.racio...@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
(*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918
~~ - (p.dalga...@biostat.ku.dk) FAX: (+45) 35327907
--
O__ Peter Dalgaard Øster Farimagsgade 5, Entr.B
c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K
(*) \(*) -- University of Copenhagen
