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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?

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Your second equation is not a proper tensor equation, as the RHS has $\alpha$ downstairs, where in the LHS it is upstairs. In order to have a correct relationship, you need to sum over an upstairs and a downstairs $\mu$, otherwise you will be left with a $\mu$ in the RHS.

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