duvin103time
New member
- Joined
- Jun 14, 2006
- Messages
- 9
Hi, every one!
I have a problem about Schmidt's ortho-normalization and vector projection as follows.
Suppose a vector set V = {v1,v2,v3}. Then we have 3*3*3=27 cases to choose 3 vectors from set V as vector s, a1 and a2. In each case, a1 and a2 are Schmidt's ortho-normalized as b1 and b2. i.e. b1=a1/|a1|, b2 = (a2-<v2,a1>a1)/|a2-<v2,a1>a1| ( when v1 and v2 are linear dependent, just set b2 = 0 )
Now I guess, the sum of square of projection of s on b1 for all the cases should be greater than that of s on b2. Numerical results of 3D or higher dimension show this may be true. I need a proof for this but I haven't got one.
Thank you!
I have a problem about Schmidt's ortho-normalization and vector projection as follows.
Suppose a vector set V = {v1,v2,v3}. Then we have 3*3*3=27 cases to choose 3 vectors from set V as vector s, a1 and a2. In each case, a1 and a2 are Schmidt's ortho-normalized as b1 and b2. i.e. b1=a1/|a1|, b2 = (a2-<v2,a1>a1)/|a2-<v2,a1>a1| ( when v1 and v2 are linear dependent, just set b2 = 0 )
Now I guess, the sum of square of projection of s on b1 for all the cases should be greater than that of s on b2. Numerical results of 3D or higher dimension show this may be true. I need a proof for this but I haven't got one.
Thank you!