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Re: LeastSquares vs. Pseudoinverse

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109403] Re: LeastSquares vs. Pseudoinverse
  • From: carlos at colorado.edu
  • Date: Sun, 25 Apr 2010 06:25:19 -0400 (EDT)
  • References: <hqu8bj$rpb$1@smc.vnet.net>

On Apr 24, 1:58 am, eric g <eric.p... at gmail.com> wrote:
> Hello Group,
> What is the difference of those when solving the problem A.x=b?
> What are the difference scenarios of one vs. the other?
> best regards,
> Eric

The distinctions are difficult to state unless one introduces the
generalized inverse in full generality.  For example, a vector x is a
LS
(least squares) solution of A x = b if and only if

            x = A{1,3} b + (I-A{1,3}) y    y=arbitrary vector   (*)

where A{1,3} is a generalized inverse of A that satisfies Penrose's
conditions 1 and 3. The pseudoinverse A{1,2,3,4} satisfies conditions
1 thru 4 so it is a  particular case of (*)    For details see the
book of
Ben-Israel and  Greville,  2nd ed, Chapters 2-3.  (In that book the
pseudoinverse is called the Moore-Penrose inverse)


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