Generalised Inverse:
The Moore-Penrose Generalisied Inverse is probably better defined as a
pseudo-Inverse that arises in solving least squares problems.
Another well known pseudo-Inverse is the so-called Drazin pseudo-Inverse.
If memory serves (and it's been 10-12 years!) it can be obtained via a
diagonalisation.
Anyway, I dare say Prof. Ripley (among others) probably has "all the
low-down" on this stuff.
Gerard
Torsten Hothorn
<Torsten.Hothorn at rzmail.uni-er To: Philippe
Grosjean <phgrosje at ulb.ac.be>
langen.de> cc: r-help at
stat.math.ethz.ch
Sent by: Subject: RE: [R]
General Matrix Inverse
owner-r-help at stat.math.ethz.ch
18/10/01 08:52
> I use solve(x) to find the inverse of a matrix (don't know what a
"general> inverse" is). By the way, what is better: solve(x), qr.solve(x) or
ginv(x)?> ginv(x) seems to give results for matrices where solve and qr.solve
return> an error:
>
> > x <- matrix(1:9, 3, 3)
> > x
> [,1] [,2] [,3]
> [1,] 1 4 7
> [2,] 2 5 8
> [3,] 3 6 9
> > solve(x)
> Error in solve.default(x) : singular matrix `x' in solve
> > qr.solve(x)
> Error in qr.solve(x) : singular matrix `x' in solve
> > ginv(x)
> [,1] [,2] [,3]
> [1,] -0.6388889 -5.555556e-02 0.5277778
> [2,] -0.1666667 4.163336e-17 0.1666667
> [3,] 0.3055556 5.555556e-02 -0.1944444
>
if A is singular, A^-1 is not defined but a generalized inverse G is,
namely
G is generalized inverse of A <=>
A G A = A (sometimes G is written as A^-)
G is not unique, but adding 3 conditions
- G A G = G
- t(G A) = G A
- t(A G) = A G
makes G unique (Moore-Penrose-Inverse)
Torsten
> Regards,
>
> Philippe Grosjean
>
>
>
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> -----Message d'origine-----
> De : owner-r-help at stat.math.ethz.ch
> [mailto:owner-r-help at stat.math.ethz.ch]De la part de Prof Brian Ripley
> Envoye : jeudi 18 octobre 2001 04:25
> A : Randall Skelton
> Cc : r-help at stat.math.ethz.ch
> Objet : Re: [R] General Matrix Inverse
>
>
> On Wed, 17 Oct 2001, Randall Skelton wrote:
>
> > What is the easiest (not the fastest) way to find the general inverse
of a> > matrix in R?
>
> If you mean the generalized inverse, ginv() in package MASS. Otherwise,
> pleae tell us what a `general inverse' is.
>
>
> --
> Brian D. Ripley, ripley at stats.ox.ac.uk
> Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/
> University of Oxford, Tel: +44 1865 272861 (self)
> 1 South Parks Road, +44 1865 272860 (secr)
> Oxford OX1 3TG, UK Fax: +44 1865 272595
>
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