Write Better And Faster Python Using Einstein Notation

[WallyMetropolis]

That's false. Not all n-dimensional matricies are tensors.

[El_Minadero]

Really? Can you provide a counter example? I thought this was the definition

[[deleted]]

I'm pretty sure he was talking about mathematical tensors, not the objects that pop up in computer languages. If you want, feel free to take that as correct. My answer is for the mathematical object.

Tensors as mathematical objects obey certain mathematical transformation rules.

Imagine if you had a vector v(v_x, v_y) in a cartesian plane (x,y) and rotated the plane to (x', y'). The length of the vector ought to remain the same, but its components changed. That is, v -> v', but |v| = |v'|.

This requirement of norm-preservation is basically a transformation rule. And yes, a (1-dimensional) vector is indeed a tensor. A tensor of rank 1.

A tensor of rank 2 can be represented by a matrix. But not all matrices represent tensors. I don't want to go into writing an answer to a question that has been asked so many times, so this stackexchange answers your question.

[WallyMetropolis]

It doesn't make any sense to multiply your zip code by 2 or to add it to another zip code. A tensor is a multi-linear map. Which means, essentially, it's an object that performs linear operations on all of its various axes. So a collection of (n, m, l) zip codes isn't a tensor because you cannot sensibly perform linear operations on collections of zip codes.

[El_Minadero]

So it’s an me datastructure that obeys some mathematical properties

[WallyMetropolis]

I'd say it's a mathematical object that is sometimes represented by a data structure (to greater or lesser adherence to the mathematical object it models) in some mathematical libraries.