I have a $1$-D matrix $V_x$ of length $N\times 1$ (here $x$ represent that the elements of $V$ depend upon variable $x$). And a 2D matrix $T$ of dimension $N\times N$. The elements of both matrices are quantum operators, so there order matters. Both matrices are connected via a relations $$ V_x^\dagger = \left(TV_{-x}\right)^T \tag{1} $$ and $$ V_{x} = T\left(V_{-x}^\dagger\right)^T \tag{2} $$ here $\dagger$ represents the conjugate transpose of the matrix.
I want to write Eq $(1)$ and $(2)$ in summation notation, something like $\left(V_x^\dagger\right)_i=\left(\sum_j {T_{ij}\left(V_{-x}\right)_j}\right)^T$.
The main confusion that I am having is about the Transpose operators. I am unsure which element should come first when written in the summation notations.
Sorry for such a messed-up question.