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).