Positive Definite Matrix

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."

-- 
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?