Is the rank of a Tensor different from the rank of a Matrix?

"number of dimensions" is a pretty terrible description. Looking at the website, their interpretation of "tensor" is not the same as the strict mathematical definition (which is to do with multilinear maps with certain properties).

As far as that program is concerned, a tensor is a vector of vectors of vectors of... of vectors, where the rank is the number of nestings of "of vectors", so

• A tensor of rank $1$ is a vector, which is a one-dimensional array, [a,b].
• A tensor of rank $2$ is a vector of vectors, or a matrix, or a two-dimensional array, [[a,b],[c,d]].
• A tensor of rank $3$ is a vector of vectors of vectors, so something with three nestings, [[[a,b],[c,d]],[[e,f],[g,h]]] sort of thing.
• &c.

The "dimension" here is the (tensorial) rank, or the number of inputs you need to locate an entry. This is of course not the same as "dimension" in the sense that something has "dimension $n$" if is an ordered list of length $n$.