A tensor may be represented as a product of vectors (arithmetic product, not cross, not dot), such as a four vector.

Where did you get that assumption from? The fact that the product of vectors is a tensor doesn't mean that the converse is true. For example; The Faraday tensor, aka the electromagnetic field tensor, is a second rank tensor. This tensor cannot be written as the product of two 4-vectors.

A vector has a magnitude (scalar).

May a tensor have a magnitude?

Not in general. You can always contract a tensor until all that's left is a scalar (i.e. a tensor of rank 0) but that won't yield a quantity that is very descriptive of the what the tensor is defined as.

Note: Unlike your high school math and science classes, the term

**scalar** is defined as

**a tensor of rank zero**. For reference please see:

http://mathworld.wolfram.com/Scalar.htmlMay it be represented as the product of vector magnitudes?

If so this may lead to interesting physics.

If you have a tensor which is defined as the product of 4-vectors then you could

*define* the magnitude as the product of the norms of the 4-vectors. But again, this may not be useful. For example; consider 10 tensors, each of which defined in terms of a number of 4-vectors, each tensor having a different number of 4-vectors. Let each 4-vector be the 4-momentum of a photon of various energies, each photon having a different energy. Then regardless of the number of 4-vectors in each tensor, the magnitude so defined will always be zero since the norm of a photons 4-momentum is zero.

Question: Do you want the magnitude to be invariant, i.e. a scalar quantity?