I think it may be important, but I am not sure. Actually I am trying to
program the adaptive nearest neighbor method proposed by Hastie and
Tibshirani. I am following the steps in the book 'The elements of
statistical learning' by Hastie, Tibshirani and Friedman, in which the local
metric is defined as W^(-1/2)[B*+I]W^(-1/2), where W is the pooled
within-class covariance matrix.
Cindy

On Mon, Aug 10, 2009 at 4:28 PM, Gabor Grothendieck <ggrothendieck@gmail.com
> wrote:

> If its not important which of many solutions you use then
> the generalized inverse can be used, say.  Just use 0
> for each small eigenvalue and 1/sqrt(x) for the others.
>
> On Mon, Aug 10, 2009 at 6:36 PM, cindy Guo<cindy.guo3@gmail.com> wrote:
> > Hi, Ted,
> >
> > Thanks for the sample code. It is exactly what I want. But can I ask
> another
> > question? The matrix for which I want the negative square root is a
> > covariance matrix. I suppose it should be positive definite, so I can do
> > 1/sqrt(V) as you wrote. But the covariance matrix I got in R using the
> > function cov has a lot of negative eigenvalues, like -5.338634e-17, so
> > 1/sqrt(V) generates NA's. Can you tell what's the problem here.
> >
> > Thanks,
> > Cindy
> >
> > On Mon, Aug 10, 2009 at 2:53 PM, Ted Harding
> > <Ted.Harding@manchester.ac.uk>wrote:
> >
> >>  On 10-Aug-09 21:31:30, cindy Guo wrote:
> >> > Hi, All,
> >> > If I  have a symmetric matrix, how can I get the negative square root
> >> > of the matrx, ie. X^(-1/2) ?
> >> >
> >> > Thanks,
> >> >
> >> > Cindy
> >>
> >>  X <- matrix(c(2,1,1,2),nrow=2)
> >>  X
> >> #      [,1] [,2]
> >> # [1,]    2    1
> >> # [2,]    1    2
> >>
> >>  E <- eigen(X)
> >>  V <- E$values
> >>  Q <- E$vectors
> >>  Y <- Q%*%diag(1/sqrt(V))%*%t(Q)
> >>  Y
> >> #            [,1]       [,2]
> >> # [1,]  0.7886751 -0.2113249
> >> # [2,] -0.2113249  0.7886751
> >>
> >>  solve(Y%*%Y)    ## i.e. find its inverse
> >> #      [,1] [,2]
> >> # [1,]    2    1
> >> # [2,]    1    2
> >>
> >> Hence (Y%*%Y)^(-1) = X, or Y = X^(-1/2)
> >>
> >> Hopingb this helps,
> >> Ted.
> >>
> >> --------------------------------------------------------------------
> >> E-Mail: (Ted Harding) <Ted.Harding@manchester.ac.uk>
> >> Fax-to-email: +44 (0)870 094 0861
> >> Date: 10-Aug-09                                       Time: 22:53:25
> >> ------------------------------ XFMail ------------------------------
> >>
> >
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> >
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> > PLEASE do read the posting guide
> http://www.R-project.org/posting-guide.html<http://www.r-project.org/posting-guide.html>
> > and provide commented, minimal, self-contained, reproducible code.
> >
>

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