CODEWITHSUNDEEP
matlabmatrixrotationtransform
Bastien Pasdeloup
Bastien Pasdeloup

Reputation: 1129

Rotate a basis to align to vector

I have a matrix M of size NxP. Every P columns are orthogonal (M is a basis). I also have a vector V of size N.

My objective is to transform the first vector of M into V and to update the others in order to conservate their orthogonality. I know that the origins of V and M are the same, so it is basically a rotation from a certain angle. I assume we can find a matrix T such that T*M = M'. However, I can't figure out the details of how to do it (with MATLAB).

Also, I know there might be an infinite number of transforms doing that, but I'd like to get the simplest one (in which others vectors of M approximately remain the same, i.e no rotation around the first vector).

A small picture to illustrate. In my actual case, N and P can be large integers (not necessarily 3):

enter image description here

Thanks in advance for your help!

[EDIT] Alternative solution to Gram-Schmidt (accepted answer)

I managed to get a correct solution by retrieving a rotation matrix R by solving an optimization problem minimizing the 2-norm between M and R*M, under the constraints:

Due to the solver constraints, I couldn't indicate that R*M[0] ... R*M[P-1] are all pairwise orthogonal (i.e (R*M)' * (R*M) = I).

Luckily, it seems that with this problem and with my solver (CVX using SDPT3), the resulting R*M[0] ... R*M[P-1] are also pairwise orthogonal.

Upvotes: 2

Views: 1845

Answers (2)

Anonymous
Anonymous

Reputation: 1968

Option # 1 : if you have some vector and after some changes you want to rotate matrix to restore its orthogonality then, I believe, this method should work for you in Matlab

http://www.mathworks.com/help/symbolic/mupad_ref/numeric-rotationmatrix.html (edit by another user: above link is broken, possible redirect: Matrix Rotations and Transformations)

If it does not, then ...

Option # 2 : I did not do this in Matlab but a part of another task was to find Eigenvalues and Eigenvectors of the matrix. To achieve this I used SVD. Part of SVD algorithm was Jacobi Rotation. It says to rotate the matrix until it is almost diagonalizable with some precision and invertible.

https://math.stackexchange.com/questions/222171/what-is-the-difference-between-diagonalization-and-orthogonal-diagonalization

Approximate algorithm of Jacobi rotation in your case should be similar to this one. I may be wrong at some point so you will need to double check this in relevant docs :

1) change values in existing vector

2) compute angle between actual and new vector

3) create rotation matrix and ...

  • put Cosine(angle) to diagonal of rotation matrix
  • put Sin(angle) to the top left corner of the matric
  • put minus -Sin(angle) to the right bottom corner of the matrix

4) multiple vector or matrix of vectors by rotation matrix in a loop until your vector matrix is invertible and diagonalizable, ability to invert can be calculated by determinant (check for singularity) and orthogonality (matrix is diagonalized) can be tested with this check - if Max value in LU matrix is less then some constant then stop rotation, at this point new matrix should contain only orthogonal vectors.

Unfortunately, I am not able to find exact pseudo code that I was referring to in the past but these links may help you to understand Jacobi Rotation :

Upvotes: 2

rlbond
rlbond

Reputation: 67779

I believe you want to use the Gram-Schmidt process here, which finds an orthogonal basis for a set of vectors. If V is not orthogonal to M[0], you can simply change M[0] to V and run Gram-Schmidt, to arrive at an orthogonal basis. If it is orthogonal to M[0], instead change another, non-orthogonal vector such as M[1] to V and swap the columns to make it first.

Mind you, the vector V needs to be in the column space of M, or you will always have a different basis than you had before.

Matlab doesn't have a built-in Gram-Schmidt command, although you can use the qr command to get an orthogonal basis. However, this won't work if you need V to be one of the vectors.

Upvotes: 3

Related Questions