Transpose

The object obtained by replacing all elements $a_{ij}$ with $a_{ji}$. For a second-Rank Tensor $a_{ij}$, the tensor transpose is simply $a_{ji}$. The matrix transpose, written ${\hbox{\sf A}}^{\rm T}$, is the Matrix obtained by exchanging A's rows and columns, and satisfies the identity

$$({\hbox{\sf A}}^{\rm T})^{-1}=({\hbox{\sf A}}^{-1})^{\rm T}.$$

The product of two transposes satisfies

$$({\hbox{\sf B}}^{\rm T}{\hbox{\sf A}}^{\rm T})_{ij} = (b^{\rm T})_{ik}(a^{\rm T})_{kj} = b_{ki}a_{jk} = a_{jk}b_{ki} = ({\hbox{\sf A}}{\hbox{\sf B}})_{ji}$$

$$= ({\hbox{\sf A}}{\hbox{\sf B}})^{\rm T}_{ij}.$$

Therefore,

$$({\hbox{\sf A}}{\hbox{\sf B}})^{\rm T} = {\hbox{\sf B}}^{\rm T}{\hbox{\sf A}}^{\rm T}.$$