TITLE: Matrix notation in tensor transformations
QUESTION [7 upvotes]: In special relativity one looks at coordinate transformations that consist of combinations of Lorentz boosts, rotations and reflections - members of the Lorentz group. Under an arbitrary transformation like that, a 4-vector $\vec{x}$ transforms as:
$$x'^\alpha = \Lambda^\alpha{}_\beta x^\beta$$
Where $\Lambda^\alpha{}_\beta$ is represents this transformation (is this a $(1,1)$ tensor itself?). This can be written in matrix form
$$x' = \Lambda x$$
Now, the basis vectors transform in another way:
$$\vec{e'_\alpha} = (\Lambda^{-1})^\beta{}_\alpha\vec{e_\beta}$$
A 1-form $\tilde{p}$ transforms like this too:
$$p'_\alpha = (\Lambda^{-1})^\beta{}_\alpha p_\beta$$
while the basis 1-forms obey
$$\omega'^\alpha = \Lambda^\alpha{}_\beta \omega^\beta$$
Anyway, for a tensor representable by a matrix, that is a $(0,2)$, $(2,0)$ or $(1,1)$ tensor, there are different transformation properties:
$$T'^{\alpha\beta} = \Lambda^\alpha{}_\gamma \Lambda^\beta{}_\delta T^{\gamma \delta}$$ 
$$T'_{\alpha\beta} = (\Lambda^{-1})^\gamma{}_\alpha (\Lambda^{-1})^\delta{}_\beta T_{\gamma \delta}$$ 
$$T'^{\alpha}{}_\beta = \Lambda^\alpha{}_\gamma (\Lambda^{-1})^\delta{}_\beta T^\gamma{} _\delta$$ 
My question is: how to translate these rules into matrix equations? I've seen formulas involving the transpose (no idea how that would even come about) of $\Lambda$ but never with an explanation where this came from. So how do I come up with the right order and transposes/inverses on the matrices?

REPLY [3 votes]: Let us rearrange your 
$$T'^{\alpha\beta} = \Lambda^\alpha{}_\gamma \Lambda^\beta{}_\delta T^{\gamma \delta},$$ 
into
$$T'^{\alpha\beta} = \Lambda^\alpha{}_\gamma T^{\gamma \delta}\Lambda^\beta{}_\delta,$$ 
and
one step more
$$T'^{\alpha\beta} = \Lambda^\alpha{}_\gamma T^{\gamma \delta}(\Lambda^{\top})_\delta{}^\beta{}.$$ 
In this last equation anyone could clearly see that to get the components of $T'$ one needs to multiply accordingly to
$$T'=\Lambda T\Lambda^{\top}.$$