# [R] Positive Definite Matrix

Prof Brian Ripley ripley at stats.ox.ac.uk
Sat Jan 29 18:17:05 CET 2011

```On Sat, 29 Jan 2011, David Winsemius wrote:

>
> 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."

But again, that is not usually what you want.  There is no guarantee
that the result is positive-definite enough that the Cholesky
decomposition will work.  Give up on Cholesky factors unless you have
a matrix you know must be symmetric and strictly positive definite,
and use the eigendecomposition instead (setting negative eigenvalues
to zero).  You can then work with the factorization to ensure that
(for example) variances are always non-negative because they are
always computed as sums of squares.

This sort of thing is done in many of the multivariate analysis
calculations in R (e.g. cmdscale) and in well-designed packages.

>
> --
> 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)
>>>>>> 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?
>>>>>
>
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