I need to calculate:
$f^{\mu'\nu'}=L^{\mu}_{\kappa}L^\nu_\lambda f^{\kappa\lambda}$
Where $L^\nu_\lambda$ is the usual Lorentz transformation matrix I thought that I just needed to do some normal matrix multiplications as the RHS terms are 4x4 matrices but the result does not agree with the solution I was given. Is there something else I need to consider In multip
What you need to compute in matrix form is $$F' = \Lambda F\Lambda^T$$ You can see that this is the case by a correct ordering of the indices in the expression you have given in the question.