It might be easier to think of it as a kind of Feynman diagram: a sum over paths. That's what matrix multiplication is, it's a kind of sum over weighted paths.

It might be easier to think of it as a kind of Feynman diagram: a sum over paths. That's what matrix multiplication is, it's a kind of sum over weighted paths. A matrix records all the ways to get from the set of "columns" to the set of "rows".