The Lorentz transformation matrix may be written in index form as Λμ ν. The transpose may be written (ΛT)μ ν=Λν μ. I want to apply this to convert the defining relation for a Lorentz transformation η=ΛTηΛ into index form. We have ηρσ=(ΛT)ρ μημνΛν σ The next step to obtain the correct result seems to be to write (ΛT)ρ μ=Λμ ρ However I don't see how this agrees with the way I have defined the transpose above. Why can we not use (ΛT)ρ μ=Λμ ρ, which seems to follow from the definition of the transpose above? Thanks for any help!