From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Newsgroups: sci.math
Subject: Re: Linear equation system question
Date: 28 Nov 1998 01:08:23 -0500

In article <73nqk6$hub@eng-ser1.erg.cuhk.edu.hk>,
Fung Wai Keung <wkfung@mae.cuhk.hk> wrote:
>Hi,
>
>	Would anyone point me to reference on how to solve a linear system
>Ax=b, where is A is a m x m matrix with rank r and r < m, x and b are m x
>1 vectors?  Moreover, A is symmetric and idempotent.  A does not have
>inverse and I also can't compute its pseudo-inverse as
>A^{+}=A^{T}*(AA^{T})^{-1} and AA^{T} or A^{T}A both equal to A itself.

(Presumably, A is a real matrix):
If the situation is as simple as you described, there is no problem:

If A is a symmetric idempotent matrix (also known as an orthoprojector)
then it is its own Moore-Penrose pseudoinverse. Why? One of the ways is to
check Moore-Penrose equations

A*X*A=A,         X*A*X=X,
(A*X)^T = A*X,   (X*A)^T = X*A

where you write X in place of A.

Another way: Tichonov regularization with parameter c>0 for the equation
A*x=b leads to  

    (A^T * A + c*I) * x = A^T * b

and for an orthoprojector A it has a solution  

    x = (1+c)^(-1) * A * b

The pseudosolution is obtained from letting c->0, and the limit is x=A*b.
If it is not the true solution then it is the least-squares solution of
the minimal length.

Hope it helps, ZVK(Slavek).