How to find rotational matrix for an n-dimensional subspace?

Is there any matlab function that returns a rotational matrix which rotates an n-dimensional subspace along an n-dimensional vector ?

[Matt J]

Rotates it by what "angle"?

[sam d]

Basically I want to rotate a subspace towards an n-dimensional vector which does not fall in the subspace. Therefore, the angle of rotation would be equal to the angle between the vector and the subspace.

[Matt J]

A 3D example would help. Which rotation matrix would you use to rotate the xy-plane toward the z-axis? Would you rotate about the x-axis, the y-axis, or a different one altogether?

[sam d]

Let the vectors are {x1;y1;0;0} and {x2;y2;0;0}. Then the subspace is the x-y plane. I need to rotate the subspace towards z-axis about a vector which is perpendicular to x, y and z axis.

[Matt J]

That's a 4D example. I requested a 3D one.

[sam d]

Let the y-axis is the subspace that has to be rotated towards z-axis. Then the axis of rotation is x-axis in this case. Basically I want the axis of rotation to be orthogonal to either of subspace and the vector.

[Matt J]

The subspace will always be of dimensions n-2 in R^n?

[sam d]

Thanks for your replies. In my case, it's an m-dimensional subspace in R^n ((n-m)>2). Let M is that subspace where an arbitrary vector v has the projection v_p (can be found using least square). I need to rotate M in such a way, so that v falls on v_p. How can I do that in Matlab ?

[Matt J]

When m>2, the options for the axis of rotation are non-unique. In particular, in my answer below

   N=null([U,v].');

will give you a matrix for N instead of a unique vector. You must decide how you want to resolve the ambiguity. The axis of rotation must be a linear combination of the columns of N.

[sam d]

Does the N also contain the axis of rotation which will cause v to fall on its least square projection on to the subspace ?

Answer 1

[Matt J]

Here's a solution assuming the subspace is of dimension n-2 in R^n, see also My Commment. The solution is not unique for dimensions n>3, but the code does return one solution.

     function [R,N]=getrotation(U,v)
     %U - a matrix of size nx(n-2) whose columns are an orthogonal basis 
     %    for a subspace of dimension n-2
     %
     %v - a given vector
     %
     %R - final rotation matrix
     %N - axis of rotation

        v=v(:)/norm(v);
        N=null([U,v].'); %axis of rotation

        q=U*(U.'*v);
          q=q/norm(q);

        A=null([N,v].');
        B=null([N,q].');

        AA=[N,v,A];
        BB=[N,q,B].';

         AA(:,3)=AA(:,3)*sign(det(BB*AA));

        R=AA*BB; %final rotation

[Matt J]

Does the N also contain the axis of rotation which will cause v to fall on its least square projection on to the subspace ?

In dimensions n>3, the choice of the axis of rotation and the choice of the point where v will fall are independent. In the code, I have made it so that v will map to and from its least squares projection, q. However, I could easily have made v map to any arbitrary point in the subspace spanned by U, with any axis of rotation selected from the null space spanned by N.