If I have the electromagnetic field tensor, then, under a Lorentz transformation:
$$F^{'}_{\mu\nu} = \Lambda_{\mu}^{\alpha} \Lambda_{\nu}^{\beta} F_{\alpha\beta} $$
I know that the Lorentz matrix is orthogonal, then:
$$\Lambda_{\mu}^{\alpha} \Lambda_{\mu}^{\beta} = \delta_{\alpha}^{\beta}$$
So, for every diagonal element with $ \mu = \nu $,
$$F^{'}_{\mu\mu} = \delta_{\alpha}^{\beta} \ F_{\alpha\beta} = F_{\alpha\alpha}$$
Is this correct? For every tensor, the diagonal elements will not change under a Lorentz transformation?