Thread: Matrix field?

There are scalar fields, vector fields, and tensor fields, so what about matrix fields? I can't find anything on them in a mathematical context. And I don't mean an algebraic field of matrices, if that applies, I mean a vector field sort of field where there's a vector assigned to each point in the space.

Originally Posted by Chemboy

There are scalar fields, vector fields, and tensor fields, so what about matrix fields? I can't find anything on them in a mathematical context. And I don't mean an algebraic field of matrices, if that applies, I mean a vector field sort of field where there's a vector assigned to each point in the space.

The situation you described sounds like a vector field, A vector field is defined at different points of space as having a certain magnitude and direction.

As to if there are matrix fields. I don't know, I have never heard of them.

Originally Posted by Chemboy

There are scalar fields, vector fields, and tensor fields, so what about matrix fields? I can't find anything on them in a mathematical context. And I don't mean an algebraic field of matrices, if that applies, I mean a vector field sort of field where there's a vector assigned to each point in the space.

If you identify matrices with linear transformations then you ought to be able to handle any idea logically relating to a "matrix field" with an appropriate tensor field constructed using tensor products of the tangent and cotangent bundles. Or, if you like you can consider the set of mxn matrices to be the Euclidean space of dimension mn and then a matrix field would just be a section of the fiber bundle with fiber .

Your post didn't make sense to me back then so I kind of gave up on it, but now that Guitarist has done some tensors it makes perfect sense.