The diagonal blocks

Let $J$ be the counteridentity matrix. That is, $(x_1,x_2,\ldots,x_m)J=(x_m,x_{m-1},\ldots,x_1)$. We note $J^2=I$.

We form the diagonal block

$$P_1=A_1+A_2+A_2^t+A_3$$

and find the structure

$$P_1= \left( \begin{array}{l\vert l} q_1 & \hat q_s \\ \hline \hat q_s^t & \tilde Q_1 \end{array}\right)$$

where $\hat q_s$ is an even row vector, or $\hat q_s J = \hat q_s$ and

$$JQ_1J=Q_1.$$

We form the diagonal block

$$P_3=A_1-A_2-A_2^t+A_3$$

and find the structure

$$P_3= \left( \begin{array}{l\vert l} q_3 & \hat q_a \\ \hline \hat q_a^t & \tilde Q_3 \end{array}\right)$$

where $\hat q_a$ is an odd row vector, or $\hat q_a J = -\hat q_a$ and

$$JQ_3J=Q_3.$$

We form the diagonal block

$$P_2=A_1+\imath (A_2+A_2^t) -A_3$$

and find the structure

$$P_2= \left( \begin{array}{l\vert l} q_2 & \hat e_2 \\ \hline \hat e_2^* & \tilde Q_2 \end{array}\right)$$

where

$$\hat e_2= \Re(\hat e_2)(I-\imath J).$$

and

$$JQ_2J=\overline{Q}_2.$$

We recall that $P_4=\overline{P}_2.$

Now form unitary matrices

$$R_1= \left( \begin{array}{ll} 1 & \\ & J \end{array}\right)$$

$$R_3= \left( \begin{array}{ll} -1 & \\ & J \end{array}\right)$$

$$R_2= \left( \begin{array}{ll} \imath & \\ & J \end{array}\right)$$

$$R_1^2=R_3^2=R_2*\overline{R}_2=I$$

From the above properties of the diagonal blocks

$$R_1P_1R_1=P_1$$

$$R_3P_3R_3=P_3$$

$$R_2P_2\overline{R}_2=\overline{P}_2$$