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

2 messages · simona.racioppi at libero.it, Peter Dalgaard

Mon, Nov 23, 2009 4:18 AM #

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?

Best,
Simon

project.org>

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

Peter Dalgaard

Mon, Nov 23, 2009 5:09 AM #

simona.racioppi at libero.it wrote:

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