[R] Positive Definite Matrix

[David Winsemius]

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.definite.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?
>
>>
>
>
> David Winsemius, MD
> West Hartford, CT
>
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David Winsemius, MD
West Hartford, CT