In Greenstadt's original paper the Weighted Frobenius norm is proposed without mention of its relation to scale-invariance. It is claimed that this norm is scale-invariant. That I take to mean for any linear transformation $T$, matrix $A$ has the same norm as matrix $TA$. I am not even sure this is is true (plug in the identity matrix for the weight matrix $W$). Perhaps I got the wrong invariance definition.
I have yet to found what exactly is this scale-invariance referring to. It's proof is supposed to be very straightforward because no one seemed to think it necessary to be written down explicitly.
Thank you so much !