Suppose I want to find the orthogonal projection of (x1,x2,y1,y2) such that x1=x2, y1=y2. I have to calculate the A matrix whose columns are the basis vectors of given subspace. I choose A=[v1;v2] as basis vector combination, where v1=[1 0 1 0] and v2=[0 1 0 1]. Then I calculated the Projection matrix as P=A(AT A)-1A.
Now if I want to find the Projection matrix of (x1,x2,x3,y1,y2,y3,z1,z2,z3) such that
(x2−x1)2+(y2−y1)2+(z2−z1)2=64
(x2−x3)2+(y2−y3)2+(z2−z3)2=36
(x3−x1)2+(y3−y2)2+(z3−z1)2=100
Do I have to find the basis vector for calculating Projection matrix? If yes, how?
Is there any other way to find its Projection matrix (P)?
Thank you for kind response. I know, I can use nonlinear optimization procedure but is it not possible to find basis vectors satisfying the given nonlinear equations and form a projection matrix? I will explain it using simple example.
If I take an example in R3, that is x^2+y^2+z^2=1. I can write [1,0,0], [0,1,0] and [0,0,1] as my basis in R3. I can write A=[1 0 0;0 1 0;0 0 1], and subsequently I can find the projection matrix using these basis,that is P=A(AT A)-1A. Is this the right way? or is there any thing wrong in it? If right, why not extend it for R9 for above non linear equations? Kindly help me out.
For curved surface there is no projection. But there is a point (points) at the surface at the small distance from the initial poyint. It is the standard variational problem (conditional minimum)
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