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)