Four vectors and Lorentz invariance

[electronman]

Does anyone know where I can find a mathematical proof that the norm of any four-vector is Lorentz invaraint?

[DW]

This sounds like a homework problem, but I am feeling generous. First verify by direct Lorentz transfomation of the special relativistic covariant metric tensor that it is unchanged in a Lorentz transformation.
Then consider the quantity \eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu}.
By the transformation property definition of a four vector:
\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = \eta' _{\mu}_{ \nu}\Lambda ^{\mu}_{ \kappa}T^{\kappa}\Lambda ^{\nu}_{ \lambda} T^{\lambda}
Regroup so as to work the transformation on the metric tensor first in the summations.
\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = (\Lambda ^{\mu}_{ \kappa}\Lambda ^{\nu}_{ \lambda}\eta' _{\mu}_{ \nu})T^{\kappa} T^{\lambda}
At this point you should have already verified the following step as I mentioned:
\eta '_{\mu}_{ \nu}T'^{\mu}T'^{\nu} = \eta _{\kappa}_{ \lambda}T^{\kappa} T^{\lambda}
QED