Positive Definite Matrix

On Sat, 29 Jan 2011, David Winsemius wrote:

On Jan 29, 2011, at 10:11 AM, David Winsemius wrote:

On Jan 29, 2011, at 9:59 AM, John Fox wrote:
Dear David and Alex,
I'd be a little careful about testing exact equality as in all(M  
== t(M) and
careful as well about a test such as all(eigen(M)$values > 0)  
since real
arithmetic on a computer can't be counted on to be exact.
Which was why I pointed to that thread from 2005 and the existing  
work that had been put into packages. If you want to substitute  
all.equal for all, there might be fewer numerical false alarms,  
but I would think there could be other potential problems that  
might deserve warnings.
In addition to the two "is." functions cited earlier there is also  
a "posdefify" function by Maechler in the sfsmisc package: "  
Description : From a matrix m, construct a "close" positive  
definite one."
But again, that is not usually what you want.  There is no guarantee  
that the result is positive-definite enough that the Cholesky  
decomposition will work.
I don't see a Cholesky decomposition method being used in that  
function. It appears to my reading to be following what would be  
called an eigendecomposition.
David.

> Give up on Cholesky factors unless you have a matrix you know must  
> be symmetric and strictly positive definite, and use the  
> eigendecomposition instead (setting negative eigenvalues to zero).   
> You can then work with the factorization to ensure that (for  
> example) variances are always non-negative because they are always  
> computed as sums of squares.
>
> This sort of thing is done in many of the multivariate analysis  
> calculations in R (e.g. cmdscale) and in well-designed packages.
>
>>
>> -- 
>> David.
>>>>> On Jan 29, 2011, at 7:58 AM, David Winsemius wrote:
>>>>>> On Jan 29, 2011, at 7:22 AM, Alex Smith wrote:
>>>>>>> Hello I am trying to determine wether a given matrix is  
>>>>>>> symmetric and
>>>>>>> positive matrix. The matrix has real valued elements.
>>>>>>> I have been reading about the cholesky method and another  
>>>>>>> method is
>>>>>>> to find the eigenvalues. I cant understand how to implement  
>>>>>>> either of
>>>>>>> the two. Can someone point me to the right direction. I have  
>>>>>>> used
>>>>>>> ?chol to see the help but if the matrix is not positive  
>>>>>>> definite it
>>>>>>> comes up as error. I know how to the get the eigenvalues but  
>>>>>>> how can
>>>>>>> I then put this into a program to check them as the just come  
>>>>>>> up with
>>>>>>> $values.
>>>>>>> Is checking that the eigenvalues are positive enough to  
>>>>>>> determine
>>>>>>> wether the matrix is positive definite?
>>>>>> That is a fairly simple linear algebra fact that googling or  
>>>>>> pulling
>>>>>> out a standard reference should have confirmed.
>>>>> Just to be clear (since on the basis of some off-line  
>>>>> communications it
>>>>> did not seem to be clear):  A real, symmetric matrix is  
>>>>> Hermitian (and
>>>>> therefore all of its eigenvalues are real). Further, it is  
>>>>> positive-
>>>>> definite if and only if its eigenvalues are all positive.
>>>>> qwe<-c(2,-1,0,-1,2,-1,0,1,2)
>>>>> q<-matrix(qwe,nrow=3)
>>>>> isPosDef <- function(M) { if ( all(M == t(M) ) ) {  # first test
>>>>> symmetric-ity
>>>>>                              if (  all(eigen(M)$values > 0) )  
>>>>> {TRUE}
>>>>>                                 else {FALSE} } #
>>>>>                              else {FALSE}  # not symmetric
>>>>>
>>>>>                        }
>>>>>> isPosDef(q)
>>>>> [1] FALSE
>>>>>>> m
>>>>>>> [,1] [,2] [,3] [,4] [,5]
>>>>>>> [1,]  1.0  0.0  0.5 -0.3  0.2
>>>>>>> [2,]  0.0  1.0  0.1  0.0  0.0
>>>>>>> [3,]  0.5  0.1  1.0  0.3  0.7
>>>>>>> [4,] -0.3  0.0  0.3  1.0  0.4
>>>>>>> [5,]  0.2  0.0  0.7  0.4  1.0
>>>>>> isPosDef(m)
>>>>> [1] TRUE
>>>>> You might want to look at prior postings by people more  
>>>>> knowledgeable than
>>>>> me:
>>>>> http://finzi.psych.upenn.edu/R/Rhelp02/archive/57794.html
>>>>> Or look at what are probably better solutions in available  
>>>>> packages:
>>>>> http://finzi.psych.upenn.edu/R/library/corpcor/html/rank.condition.html
>>>>> http://finzi.psych.upenn.edu/R/library/matrixcalc/html/is.positive.definit
>>>>> e.html
>>>>> --
>>>>> David.
>>>>>>> this is the matrix that I know is positive definite.
>>>>>>> eigen(m)
>>>>>>> $values
>>>>>>> [1] 2.0654025 1.3391291 1.0027378 0.3956079 0.1971228
>>>>>>> $vectors
>>>>>>>       [,1]        [,2]         [,3]        [,4]        [,5]
>>>>>>> [1,] -0.32843233  0.69840166  0.080549876  0.44379474   
>>>>>>> 0.44824689
>>>>>>> [2,] -0.06080335  0.03564769 -0.993062427 -0.01474690   
>>>>>>> 0.09296096
>>>>>>> [3,] -0.64780034  0.12089168 -0.027187620  0.08912912  
>>>>>>> -0.74636235
>>>>>>> [4,] -0.31765040 -0.68827876  0.007856812  0.60775962   
>>>>>>> 0.23651023
>>>>>>> [5,] -0.60653780 -0.15040584  0.080856897 -0.65231358   
>>>>>>> 0.42123526
>>>>>>> and this are the eigenvalues and eigenvectors.
>>>>>>> I thought of using
>>>>>>> eigen(m,only.values=T)
>>>>>>> $values
>>>>>>> [1] 2.0654025 1.3391291 1.0027378 0.3956079 0.1971228
>>>>>>> $vectors
>>>>>>> NULL
>>>>>>> m <- matrix(scan(textConnection("
>>>>>> 1.0  0.0  0.5 -0.3  0.2
>>>>>> 0.0  1.0  0.1  0.0  0.0
>>>>>> 0.5  0.1  1.0  0.3  0.7
>>>>>> -0.3  0.0  0.3  1.0  0.4
>>>>>> 0.2  0.0  0.7  0.4  1.0
>>>>>> ")), 5, byrow=TRUE)
>>>>>> #Read 25 items
>>>>>>> m
>>>>>> [,1] [,2] [,3] [,4] [,5]
>>>>>> [1,]  1.0  0.0  0.5 -0.3  0.2
>>>>>> [2,]  0.0  1.0  0.1  0.0  0.0
>>>>>> [3,]  0.5  0.1  1.0  0.3  0.7
>>>>>> [4,] -0.3  0.0  0.3  1.0  0.4
>>>>>> [5,]  0.2  0.0  0.7  0.4  1.0
>>>>>> all( eigen(m)$values >0 )
>>>>>> #[1] TRUE
>>>>>>> Then i thought of using logical expression to determine if  
>>>>>>> there are
>>>>>>> negative eigenvalues but couldnt work. I dont know what error  
>>>>>>> this is
>>>>>>> b<-(a<0)
>>>>>>> Error: (list) object cannot be coerced to type 'double'
>>>>>> ??? where did "a" and "b" come from?
>>
>> ______________________________________________
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>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>
> -- 
> Brian D. Ripley,                  ripley at stats.ox.ac.uk
> Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
> University of Oxford,             Tel:  +44 1865 272861 (self)
> 1 South Parks Road,                     +44 1865 272866 (PA)
> Oxford OX1 3TG, UK                Fax:  +44 1865 272595

David Winsemius, MD
West Hartford, CT

Positive Definite Matrix

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