Positive Definite Matrix
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
>
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
David Winsemius, MD
West Hartford, CT