Projection of Ax on By where norm(x)=norm(y)=1 and A,B are matrices.

I know how to project a vector x on vector y. But I would like to know if there is a function that could help me by projecting on where A and B are matrices and ∀ .

In other words, how can I get vectors of norm 1.

Answer 1

[Torsten]

The projection of A*x on B*y is

dot(A*x,B*y)/dot(B*y,B*y)*B*y

I don't know where you want norm = 1 come into play here.

[Ahsan Razaq]

I want to check the projection for every x and y such that norms of x and y are 1.

The question is more related towards finding all values of vectors x and y such that their norms are 1.

I believe I can write a routine to check the projections.

So, for a vector of length four x=[1 0 0 0],[0 1 0 0], [0 0 0 1] and [.5 .5 .5 .5]. all have norm 1.

[Torsten]

The usual definition of "projection of a vector Ax on a vector By" means that the result is independent of the length of By and thus the length of y:

Ax = [2 2];
By = [1 0];
dot(Ax,By)/dot(By,By)*By

ans = 1×2

2 0

By = [5 0];
dot(Ax,By)/dot(By,By)*By

ans = 1×2

2 0

Is this also true in your case ?

[Ahsan Razaq]

Yes, it is true which is why I only take the eigenvectors of B for my study.

[Torsten]

So say you can randomly generate vectors x and y of norm 1 uniformly distributed on the surface of the sphere with norm 1, you calculate the projection of A*x on B*y and get a vector v(Ax,By). How do you want to deal with all these projection vectors ?

[Ahsan Razaq]

I want the magnitude of projection of A*x on y (say y eigenvector of B), to be less than a threshold for all norm(x)=1.

[Torsten]

Should this be a constraint in an optimization problem ?

[Ahsan Razaq]

For example.

B=[0.5609 -0.0210 0.3960 0.0641;

0.4298 0.8544 -0.0977 -0.1864;

0.2962 0.0047 0.2928 0.4064;

-0.3366 0.1846 -0.0148 1.1669];

has eigenvector y=[-0.0748 -0.2735 0.2357 0.9295];

and I have A=diag([0.9597 0.2763 0.1839 0.9323];

Then the magnitude of projection of A*[0 0 0 1]' on y is .8666 which is a value. In this case the threshold is .76.

In the example I chose x=[0 0 0 1], I want to know how I can code x such that it has norm 1.

I know there are infinite number of vectors with norm 1, but I can have samples say about 100.

[Torsten]

Npoints = 100;
Ndim = 4; % points in a circle. ergo, easy to plot.
x = randn(Npoints,Ndim); % A normal dist is symmetrical
x = x./sqrt(sum(X.^2,2)); % project to the surface of the Ndim-sphere
x

[Ahsan Razaq]

Good Thanks.