[westurner]

AFAIU, Matrices are categorically subsets of Tensors where the product operator, at least, is not the tensor product but the Dot product.

Dot product: https://en.wikipedia.org/wiki/Dot_product

Matrix multiplication > Dot product, bilinear form and inner product: https://en.wikipedia.org/wiki/Matrix_multiplication#Dot_prod...

> The dot product of two column vectors is the matrix product

Tensor > Geometric objects https://en.wikipedia.org/wiki/Tensor :

> The transformation law for a tensor behaves as a functor on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as local diffeomorphisms.) This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes.[24] Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles.[25][26]

[zarzavat]

Mathematicians are comfortable with the idea that a matrix can be over any set, in other words matrix is a mapping from the cartesian product of two intervals [0,N_i) to a given set S. Of course to define matrix multiplication you need at least a ring, but a matrix doesn’t have to define such an operation, and there are many alternative matrix products, for example you can define the Kronecker product with just a matrix over a group. Or no product at all for example a matrix over the set {Red,Yellow,Blue}.

Tensors require some algebraic structure, usually a vector space.