# Transpose of (1,1) tensor

When we transpose a (1,1) tensor, shall we simply switch the two indices while keeping their upper/lower positions or switch them and also switch their upper/lower positions? In general, would the left/right order matter for a tensor? Is it true that in contracting indices between two tensors, we want the contracted index to be right close to each other?

1. If there is only Grassmann-even variables, then the transposed tensor is $$\tag{1} (A^{T})_j{}^i ~:=~ A^i{}_j$$ in local coordinates.
• Are you sure the tranpose of a tensor is as (1)? If A is lorentz transformation and B is its inverse, then $B_j{}^i=A^i{}_j$ , which is certainly not A's transpose. – Shadumu Mar 13 '15 at 1:09
• A Lorentz matrix satisfies $\Lambda^T\eta\Lambda=\eta$. Or equivalently $\Lambda^T=\eta\Lambda^{-1}\eta^{-1}$. Or equivalently with indices: $\Lambda^{\sigma}{}_{\mu}$$=(\Lambda^T)_{\mu}{}^{\sigma}$$=\eta_{\mu\nu}(\Lambda^{-1})^{\nu}{}_{\rho}\eta^{\rho\sigma}$$=(\Lambda^{-1})_{\mu}{}^{\sigma}$. We stress that the last eq. does not imply that $\Lambda^T$ and $\Lambda^{-1}$ are the same matrix, cf. e.g. this Phys.SE post. – Qmechanic Mar 13 '15 at 19:37
• @Qmechanic♦ I know that it has to be wrong but...Why your last equation does not imply that $\Lambda^T=\Lambda^{-1}$ – MattG88 Jul 20 '18 at 23:30