I think it will serve better to think about it as a difference of emphasis, rather than a difference of category. Maybe the term has been introduced to make the

I think it will serve better to think about it as a difference of emphasis, rather than a difference of category. Maybe the term has been introduced to make the subject seem more tangible to newcomers? I notice that University of Missouri doesn't also have a "linear algebra" course. As you say, "Matrix Theory is about matrices as a data structure" ends up being a pretty hollow concept anyway, because the important part is always related to matrix multiplication. The example of Markov chain models only underlines that point, since the main results of that theory depend entirely on the transition function being linear, and not at all on whether we represent it by a matrix. To put it another way: matrix multiplication is composition of linear operators. You can't extricate the matrix-as-data concept from the other.