Time-reversal, and signature or simplex

For operators $\hat{\cal{O}}$, for which both Eq. (11) and (18) hold, matrix ${\cal{O}}$ in Eq. (13) can be additionally simplified, and reads

$$\left(\begin{array}{cc} A & 0 \\ 0 & \epsilon_T A^* \en... ...rray}{cc} 0 & Y \\ -\epsilon_T Y^* & 0 \end{array} \right)$$ (26)

for $\epsilon_X$=+1 and $\epsilon_X$=-1, respectively, with A hermitian, and Y antisymmetric ( $\epsilon_T$=+1) or symmetric ( $\epsilon_T$=-1). Of course, this case is identical to that described in Sec. 3.1.5, because whenever the time-reversal, and signature or simplex are conserved, the corresponding T-signature or T-simplex are also conserved, and $\epsilon_X$= $\epsilon_Z\epsilon_T$. Therefore, we may now use two different bases, and obtain two different forms of the matrix ${\cal{O}}$, (23) or (26), which lead either to real, or to block-diagonal matrices. Note that in order to diagonalize matrix ${\cal{O}}$ for $\epsilon_X$=+1, one has to diagonalize only its hermitian submatrix A, which has dimension twice smaller than ${\cal{O}}$, similarly as in Eq. (25).