[R] Positive Definite Matrix

[David Winsemius]

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.

>
> 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
>
> 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