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)