Do I need a tensor of fourth order to transform a matrix to a matrix?

Question

To transform a vector $W_j$ to another vector $V_i$ one needs a matrix (tensor of 2nd order): $$ V_i = T_{ij}W_j $$ But if I want to transform a matrix $B_{kl}$ to a matrix $A_{ij}$ do I then need a tensor of fourth order? $$ A_{ij} = T_{ijkl}B_{kl} $$ And what is the difference of this operation to a matrix multiplication? $$ A_{ij} = T_{il}A_{lj} $$ I'm a physicist, so my wording and notation (I'm using Einstein summation) might not be precise in a mathematical sense, but still hope it is clear what I mean.

Answer 1

It all depends on what you want to do. An $m \times n$ matrix viewed in a different way is a vector of dimension $mn$. A tensor of fourth order is a way of representing an arbitrary linear transform on this vector space of matrices.

On the other hand, multiplication by a matrix represents a more specific type of linear transformation on the vector space of matrices. (Recall that multiplying two matrices $A$ and $B$ is equivalent to composing the corresponding linear transformations.)