What is the suitable representation of a linear operator of matrices?

[mrezamm]

Hi there,

As you know, we can represent a Linear vector operator as a matrix product, i.e., if T(u) = v, there is a matrix A that u = A.v.

What about a linear operator of matrices. I have a T(X) = b where X belongs to R^n_1Xn_2 and b belongs to R^p. What is a suitable representation of this operator? Is this tensor or Kronecker product?

Best wishes,
Reza

 

[chiro]

Hey mrezamm and welcome to the forums.

The tensor product of linear operators (matrices) is itself a matrix. Remember that all linear operators are matrices and all tensors are multi-linear objects with their own identities (which can be composed through multiple linear objects).

The first thing you have to do to find the basis for this product of spaces is to do the same kind of thing you do for finding a basis for a set of vectors: you need to find the minimum representation for the basis by removing all the redundant linearly dependent stuff and then construct a final basis operator from this.

To aid this discussion, the best thing I think for you to do is to explicitly state the operators for each space in terms of the matrix itself and if there are any dependencies between entries of the operators for each space then state those. If both spaces are completely independent, this will simplify things greatly.