Gram-Schmidt Orthogonalization
From: Noman (nasimulnoman_at_hotmail.com)
Date: 06/11/04
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Date: Fri, 11 Jun 2004 13:03:21 +0000 (UTC)
Hello Guys,
I am confused about the solution of the following problem. Can any
body help me?
Problem:
|P1> and |P2> are two arbitrary vectors of dimension n. |e1> is
defined as
|e1> = (|P2> - |P1>) / ||(|P2> - |P1>)||
now determine vectors |e2>, |e3>, ..............|en> which are the
orthogonal basis
vectors spanning the subspace perpendicular to vector |e1>.
I am using Gram-Schmidt method for solving this problem
Solution:
Create a basis for the n dimensional space {|x1>,|x2>,|x3>,.....,|Xn>}
where |x1> = |e1>, |x2> = [0 1 0 0...0], |x3> = [0 0 1 0 ..0] ....
|xn> = [0 0 0 0....0 1]
Then I transform it into an orthogonal basis
{|y1>,|y2>,|y3>,..........,|yn>} as follows
1. |y1> = |e1>
2. |y2 > = |x2> - [{|x2>.|y1>}/ ||y1||^2 ] |y1>
3. |y3 > = |x3> - [{|x3>.|y1>}/ ||y1||^2 ] |y1> - [{|x3>.|y2>}/
||y2||^2 ] |y2>
4. |y4 > = |x4> - [{|x4>.|y1>}/ ||y1||^2 ] |y1> - [{|x4>.|y2>}/
||y2||^2 ] |y2> - [{|x4>.|y3>}/ ||y3||^2 ] |y3>
................
................
n-1
5. |yn > = |xn> - Sum [{|xn>.|yi>}/ ||yi||^2 ] |yi>
i=1
Anybody please confirm me whether this calculation is correct? If not
then please
help me with the correct calculation.
And if anybody knows any Java class for calculating the orthogonal
basis then please tell me.
Thanks.
Noman
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