[R] Positive Definite Matrix
David Winsemius
dwinsemius at comcast.net
Sat Jan 29 13:58:12 CET 2011
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
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