It's not perfectly clear whether "dyads or matrices" means "dyads, which we also call matrices" (wrong!) or "either dyads or matrices" (right, kinda!), which is

It's not perfectly clear whether "dyads or matrices" means "dyads, which we also call matrices" (wrong!) or "either dyads or matrices" (right, kinda!), which is why I picked on a slightly later bit of the document to support my accusation. It's also not perfectly clear what the author really means by saying that not all matrices "are" tensors but all tensors "are" matrices; this is part of what he says is summarizing previous remarks but I don't see any previous remarks addressing the point. The usual answer is to say that a tensor is a matrix that "transforms correctly"; in something closer to puremathematicianese, what this really means is that we're looking not exactly at matrices but at functions from coordinate systems to matrices (i.e., you plug in the coordinate system you want to use, and out come the components of the thing-that-might-be-a-tensor as a 3x3 array), and that what you've got is a tensor of a particular kind if and only if the way in which these numbers change as the coordinate system does is right. But at this point in the document he hasn't said anything remotely like that. He does eventually mention it but not until the very end, where he seems to think he's getting it as a consequence of some other definition -- but unless that definition is the (definitely wrong) one that says the only tensors are tensor products of vectors, it's not clear what it is. (It is in fact possible to define tensors in another way that's neater and coordinate-free, and derive the transformation rules as theorems. One way to do this is closely related to what this document does, but instead of (wrongly) saying that all tensors are tensor products of vectors ("dyads", etc.) what you (rightly) say is that all tensors are linear combinations of tensor products of vectors. Er, and their duals. And you actually have to say what all that means, and it turns out to be a rather elegant bit of theory. Perhaps that's what this document is gesturing towards, but it really doesn't do more than gesture.)