Einstein summation for matrix multiplication isn’t really a great example because we have a perfectly good way to describe matrix multiplication without loops or sufficies or anything else:
C = AB
Rate is multiplication (and matrix-vector multiplication) is easy to denote. Tensor multiplication (well really tensor contraction) is annoying to denote and that is the point of summation notation.
Notation is easier when there is less to denote:
- Matrix multiplication: only one thing you can do (well you can add to but this distributes)
- Tensor multiplication: all you can do is add, tensor product, and contract. First two are easy, last one you add extra notation for. This notation can also express permuting indices but this is the same as multiplication by a particular tensor and contraction so ok to do.
- map,reduce,fold,etc. These can do quite a lot but not everything (eg processing multiple and in particular different data structures is often annoying. Processing things out of the usual order is often annoying. Processing eg consecutive elements is often annoying). One gets convenience for common things by making it impossible/hard to express uncommon things. If you see something that looks like map(f,x), you know the size of the input is the same as the size of the output and (hopefully) f acts independently on each element.
- loops/tail recursion these give you more power to do things so looking at a loop there are many things that could possible happen and one must look carefully to find which.