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 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).