Conditions for precoding matrix to preserve complex conjugate symmetry on DFT vector
I think that the entries in your matrix $\bf{T}$ must obey $a_{N-n+1,N-m+1} = a^*_{n,m}$.
This is saying that the entries in row $N-n+1$ are the same as the coefficients in row n but where the coefficients are conjugated and reversed. The pattern in $\bf{T}$ for $N=4$ is

$T_4 = \left[ \begin{smallmatrix} a_{11}&amp;a_{12}&amp;a_{13}&amp;a_{14}\\ a_{21}&amp;a_{22}&amp;a_{23}&amp;a_{24} \\ a^*_{24}&amp;a^*_{23}&amp;a^*_{22}&amp;a^*_{21} \\ a^*_{14}&amp;a^*_{13}&amp;a^*_{12}&amp;a^*_{11}\end{smallmatrix} \right]$ 

I'm sure someone will come up with a better and more precise answer.