[R] General Matrix Inverse

Thu Oct 18 13:45:35 CEST 2001

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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