Suppose there is a DFT vector $\mathbf{X}$ (complex vector) with length N, which presents complex conjugate symmetry around its middle point, i.e., $X(1) = X(N-1)^*$, $X(2) = X(N - 2)^*$ and so forth. X(0) and X(N/2) are the DC and Nyquist frequency respectively, therefore are real numbers. The remaining elements are complex.
Now, suppose there is a matrix $\mathbf{T}$, with size $N \times N$, which multiplies vector X.
\begin{align} \mathbf{Y} = \mathbf{T}\mathbf{X} \end{align}
The question is:
In what conditions, for matrix $\mathbf{T}$, the complex conjugte symmetry around the middle point of the resulting vector $\mathbf{Y}$ is preserved?