# Re: [R] Computing generalized eigenvalues

From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>
Date: Fri 17 Jun 2005 - 19:48:46 EST

Prof Brian Ripley <ripley@stats.ox.ac.uk> writes:

> On Thu, 16 Jun 2005, Joshua Gilbert wrote:
>
> > I need to compute generalized eigenvalues. The eigen function in base
> > doesn't do it and I can't find a package that does.
>
> They are very rarely used in statistics, so this is not surprising.

An aside, going a bit off-topic:

However, there's the related generalized singular value decomposition:

K = U Sigma inv(T)
L = V M inv(T)

U'U = V'V = I ; Sigma and M diagonal (Sigma^2 + M^2 = I by convention) ; T regular

(Look at K'K and L'L to see the connection.)

This is used in Tikhonov regularization, which is penalized least squares, which is a statistical issue (whether numerical analysts realize it or not). Smoothing splines is a special case. Deconvolution is another.

> I presume you mean solving Ax = lambda B x: if B is non-singular this
> reduces to a conventional eigenproblem for B^{-1}A.

(There are some complications if both A and B are singular. That's why the GSVD has that peculiar-looking convention.)

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