I'm looking for the dimension of some tensor $T$ that transforms $a \times b$ matrices to $c \times d$ matrices. I initially thought it would have to be of a greater dimension, perhaps some shape like $a\times b\times c \times d$ since transforming m vector to a n vector would require a $n \times m$ matrix.
However, I can't seem to think how it would look like aside from doing the normal matrix multiplication but by dotting matrices instead of vectors.
What would be the dimensions of this $a \times b$ to $c \times d$ transform look like?