Is there a representation for tensors as multidimensional arrays that is best for a general product definition?
If $\bf v$ is a column vector and $\bf w$ is a covector (row vector), then $\bf w v$ is a scalar, but $\bf v w$ is a matrix. Is there an general operation and representation for tensors unifies all instances of products? That is, what would the tensor representation and product operation need to be for the above to hold, $A\bf v$ be a column vector, and $AB$ be a matrix, for matrices $A,B$.
Using a tuples of tuples of ... representation, would there also need to be information about whether a tuple is a row or column to define such an operation?