MATLAB, Numpy, Julia, and other scientific computing systems make heavy use of element-wise multiplication and shape-dependent addition. In this way, you can have a notion of adding two multilinear maps of different types. However, I have never seen any literature detailing the properties of such an algebra. My question is the following.
What is the algebraic structure of MATLAB based on Hadamard multiplication, and shape-dependent addition?
As far as I know, both operations are not true linear algebra since the interpretation of the operations are basis-dependent. Are there any mathematics references detailing the properties of MATLAB-like algebraic operations, and that do justice to common operations seen in MATLAB codes? What theorems can I prove about it?