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From: samah karim on 26 Mar 2010 05:39
i coded the process in MATLAB, it goes as follows:
function [F,G]=GramSchmidt(E,tol)
% Input: E is an m by n matrix of full rank m<=n
% tol: our zero tolerance
% Output: An orthogonal matrix F, the product of the Gram Schmidt process
% applied on the columns of E
% F must verify F*F'=I
[m,n]=size(E);
F=zeros(m);
G=zeros(m);
% Start the process
e=E(:,1);c=sqrt(e'*e);
f=(1/c)*e;
F(:,1)=f;
G(1,1)=c;
for k=2:m
% select the kth column of E
e=E(:,k);
% Project e on the span of f1,...f_{k-1}
p=zeros(m,1);
for j=1:k-1
G(j,k)=e'*F(:,j);
p=p+G(j,k)*F(:,j);
end
v=e-p;c=sqrt(v'*v);
if c<tol
fprintf('Process breaks at %i/n ',k)
break
else
F(:,k)=(1/c)*v;
G(k,k)=c;
end
end

it's working perfectly with real matrices, but when E is complex, the output matrix F is no longer orthogonal. The vectors F(:,j) have norms = 1 so that part is fine, but they're not mutually orthogonal. and i can't figure out the source of the problem. help please
From: Bruno Luong on 26 Mar 2010 05:49

> G(j,k)=e'*F(:,j);

Replace with
G(j,k)=F(:,j)'*e;

Bruno
From: samah karim on 31 Mar 2010 10:20
Thank u Bruno! It worked :D
But i can't figure out why t worked, why should i change it if it's complex?
Well i read it in many books that the projection should be given by e'*f?
Please help.
thank you.
From: Bruno Luong on 31 Mar 2010 10:34
"samah karim" <sammouha_007(a)hotmail.com> wrote in message <hovln3$4dr$1(a)fred.mathworks.com>...
> Thank u Bruno! It worked :D
> But i can't figure out why t worked, why should i change it if it's complex?

Because the first version of your code is the conjugate of the right expression. For real matrix, conjugate does not matter.

> Well i read it in many books that the projection should be given by e'*f?

The book is wrong then.

Bruno
From: samah karim on 31 Mar 2010 11:05
i mean i read that the projection of the 2nd column vector on the normalized 1st one, is given by e'*f
but i never found any description of the problem when the matrix is complex.
Do u have any references(websites,books) about this topic in particular(i.e. Gram-Scmidt with complex matrices) so i can understand the math behind this change?
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