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As known, any Lorentz transformation matrix [tex]\Lambda[/tex] must obey the relation [tex]\Lambda^\mu~_\nu\Lambda^\rho~_\sigma g_{\mu \rho}=g_{\nu \sigma}[/tex]. The same holds also for the inverse metric tensor [tex]g^{\nu \sigma}[/tex] which has the same components as the metric tensor itself (don't really understand why every tex formula starts from a new line), i.e. [tex]\Lambda^\mu~_\nu\Lambda^\rho~_\sigma g^{\nu \sigma}=g^{\mu \rho}[/tex]. Putting this all as a matrix relation, these two formulas are [tex]\Lambda^T~g~\Lambda=g,~~~\Lambda~g~\Lambda^T=g~~~~~(1)[/tex], where g is the metric tensor (and also the inverse metric tensor, as they are both the same). From here one can deduce that [tex]\Lambda^T=\pm\Lambda[/tex], so Lorentz transformation matrix should be either symmetric or antisymmetric. And Everything was great until today, when in Weinberg's book on quantum field theory (vol.1, formula 2.5.26, http://www.scribd.com/doc/3082871/Steven-Weinberg-The-Quantum-Theory-of-Fields-Vol-1-Foundations [Broken] , page 70, though it isn't much important) I met a Lorentz transformation matrix which is "almost" antisymmetric (it is antisymmetric, except for there aren't zero's on main diagonal).

So I guess I'm wrong somewhere. Isn't the Lorentz transformation matrix restricted to be either symmetric or antisymmetric? Or the equations (1) have other solutions too?

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# The Lorentz transformation matrix properties

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