Appendix
1.1 The proof of Lemma 1
We use induction to show (2.2). For \(k=1\), we can write
$$\begin{aligned}&\sum ^\theta _{i=1}\langle S_i(1), X_i^*-X_i(1)\rangle = \sum ^\theta _{i=1}\Big \langle A_{1,i}^HR_{1,i}(1)B_{1,i}^H, X_i^*-X_i(1)\Big \rangle \nonumber \\&\qquad + \sum ^\theta _{i=1}\Big \langle C_{1,i-1}^HR_{1,i-1}(1)D_{1,i-1}^H, X_i^*-X_i(1)\Big \rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\Big \langle A_{2,i}^HR_{2,i}(1)B_{2,i}^H, X_i^*-X_i(1)\Big \rangle \nonumber \\&\qquad + \sum ^\theta _{i=1}\Big \langle C_{2,i-1}^HR_{2,i-1}(1)D_{2,i-1}^H, X_i^*-X_i(1)\Big \rangle \nonumber \\&\quad = \sum ^\theta _{i=1}\Big \langle R_{1,i}(1), A_{1,i}(X_i^*-X_i(1))B_{1,i}\Big \rangle \nonumber \\&\qquad + \sum ^\theta _{i=1}\langle R_{1,i}(1), C_{1,i}(X_{i+1}^*-X_{i+1}(1))D_{1,i}\rangle \nonumber \\&\qquad + \sum ^\theta _{i=1}\Big \langle R_{2,i}(1), A_{2,i}( X_i^*-X_i(1))B_{2,i}\Big \rangle \nonumber \\&\qquad + \sum ^\theta _{i=1}\Big \langle R_{2,i}(1), C_{2,i}(X_{i+1}^*-X_{i+1}(1))D_{2,i}\Big \rangle \nonumber \\&\quad = \sum ^\theta _{i=1}\Big \langle R_{1,i}(1), A_{1,i} X_i^*B_{1,i} +C_{1,i}X_{i+1}^*D_{1,i}\nonumber \\&\qquad - A_{1,i} X_i(1) B_{1,i} -C_{1,i}X_{i+1}(1) D_{1,i}\Big \rangle \nonumber \\&\qquad + \sum ^\theta _{i=1}\Big \langle R_{2,i}(1), A_{2,i} X_i^*B_{2,i} +C_{2,i}X_{i+1}^*D_{2,i}\nonumber \\&\qquad - A_{2,i} X_i(1) B_{2,i} -C_{2,i}X_{i+1}(1) D_{2,i} \Big \rangle \nonumber \\&\quad = \sum ^\theta _{i=1}\Big \langle R_{1,i}(1), E_{1,i} - A_{1,i} X_i(1) B_{1,i} -C_{1,i}X_{i+1}(1) D_{1,i} \Big \rangle \nonumber \\&\qquad + \sum ^\theta _{i=1}\Big \langle R_{2,i}(1), E_{2,i} - A_{2,i} X_i(1) B_{2,i} -C_{2,i}X_{i+1}(1) D_{2,i} \Big \rangle \nonumber \\&\quad = \sum ^\theta _{i=1}\left[ \Big \langle R_{1,i}(1), R_{1,i}(1) \rangle + \langle R_{2,i}(1), R_{2,i}(1) \Big \rangle \right] \nonumber \\&\quad =\sum ^\theta _{i=1}\left[ ||R_{1,i}(1)||^2+||R_{2,i}(1)||^2\right] . \end{aligned}$$
(5.1)
Suppose that the conclusion (2.2) holds for \(k=s\). When \(k=s+1\), we obtain
$$\begin{aligned}&\sum ^\theta _{i=1}\Big \langle S_i(s+1), X_i^*-X_i(s+1)\Big \rangle =\sum ^\theta _{i=1}\Big \langle A_{1,i}^HR_{1,i}(s+1)B_{1,i}^H, X_i^*-X_i(s+1)\Big \rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\Big \langle C_{1,i-1}^HR_{1,i-1}(s+1)D_{1,i-1}^H, X_i^*-X_i(s+1)\Big \rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\Big \langle A_{2,i}^HR_{2,i}(s+1)B_{2,i}^H, X_i^*-X_i(s+1)\Big \rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\Big \langle C_{2,i-1}^HR_{2,i-1}(s+1)D_{2,i-1}^H, X_i^*-X_i(s+1)\Big \rangle \nonumber \\&\qquad +\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(s+1)||^2+||R_{2,j}(s+1)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(s)||^2+||R_{2,j}(s)||^2\right] }\nonumber \\&\qquad \times \sum ^\theta _{i=1}\Big \langle S_i(s), X_i^*-X_i(s+1)\Big \rangle \nonumber \\&\quad =\sum ^\theta _{i=1}\Big \langle R_{1,i}(s+1), A_{1,i} X_i^*B_{1,i}\nonumber \\&\qquad +\,C_{1,i}X_{i+1}^*D_{1,i} - A_{1,i} X_i(s+1) B_{1,i}\nonumber \\&\qquad -\,C_{1,i}X_{i+1}(s+1) D_{1,i} \Big \rangle \nonumber \\&\qquad +\, \sum ^\theta _{i=1}\Big \langle R_{2,i}(s+1), A_{2,i} X_i^*B_{2,i} +C_{2,i}X_{i+1}^*D_{2,i}\nonumber \\&\qquad - \,A_{2,i} X_i(s+1) B_{2,i}-\,C_{2,i}X_{i+1}(s+1) D_{2,i} \Big \rangle \nonumber \\&\qquad +\,\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(s+1)||^2+||R_{2,j}(s+1)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(s)||^2+||R_{2,j}(s)||^2\right] }\nonumber \\&\qquad \times \sum ^\theta _{i=1}\Big \langle S_i(s), X_i^*-X_i(s)\nonumber \\&\qquad -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(s)||^2+||R_{2,j}(s)||^2\right] }{\sum ^\theta _{j=1}||S_j(s)||^2}S_i(s)\Big \rangle \nonumber \\&\quad =\sum ^\theta _{i=1}\left[ ||R_{1,i}(s+1)||^2+||R_{2,i}(s+1)||^2\right] \nonumber \\&\qquad +\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(s+1)||^2+||R_{2,j}(s+1)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(s)||^2+||R_{2,j}(s)||^2\right] }\nonumber \\&\qquad \times \sum ^\theta _{i=1}\langle S_i(s), X_i^*-X_i(s)\rangle \nonumber \\&\qquad -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(s+1)||^2+||R_{2,j}(s+1)||^2\right] }{\sum ^\theta _{j=1}||S_j(s)||^2}\sum ^\theta _{i=1}\langle S_i(s), S_i(s)\rangle \nonumber \\&\quad =\sum ^\theta _{i=1}\left[ ||R_{1,i}(s+1)||^2+||R_{2,i}(s+1)||^2\right] . \end{aligned}$$
(5.2)
Therefore, the conclusion (2.2) holds by the principle of induction.
1.2 The proof of Lemma 2
Step I Noting that the inner product is commutative, it is enough to prove Lemma 2 for \(1\le u< v\le r.\) For \(u=1\) and \(v=2\), we can write
$$\begin{aligned}&\sum ^\theta _{i=1}\left[ \Big \langle R_{1,i}(1), R_{1,i}(2)\Big \rangle +\Big \langle R_{2,i}(1), R_{2,i}(2)\Big \rangle \right] \nonumber \\&\quad =\sum ^\theta _{i=1}\langle R_{1,i}(1), R_{1,i}(1)-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }{\sum ^\theta _{j=1}||S_j(1)||^2}\nonumber \\&\qquad \times \left[ A_{1,i}S_i(1)B_{1,i}+C_{1,i}S_{i+1}(1)D_{1,i}\right] \rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\langle R_{2,i}(1), R_{2,i}(1)-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }{\sum ^\theta _{j=1}||S_j(1)||^2}\nonumber \\&\qquad \times \left[ A_{2,i}S_i(1)B_{2,i}+C_{2,i}S_{i+1}(1)D_{2,i}\right] \rangle \nonumber \\&\quad =\sum ^\theta _{i=1}\left[ ||R_{1,i}(1)||+||R_{2,i}(1)||\right] -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }{\sum ^\theta _{j=1}||S_j(1)||^2}\nonumber \\&\qquad \times \Bigg [ \sum ^\theta _{i=1}\langle R_{1,i}(1),A_{1,i}S_i(1)B_{1,i}+C_{1,i}S_{i+1}(1)D_{1,i}\rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\langle R_{2,i}(1), A_{2,i}S_i(1)B_{2,i}+C_{2,i}S_{i+1}(1)D_{2,i}\rangle \Bigg ]\nonumber \\&\quad =\sum ^\theta _{i=1}\left[ ||R_{1,i}(1)||+||R_{2,i}(1)||\right] -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }{\sum ^\theta _{j=1}||S_j(1)||^2}\nonumber \\&\qquad \times \Bigg [ \sum ^\theta _{i=1}\langle A_{1,i}^HR_{1,i}(1)B_{1,i}^H,S_i(1)\rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\langle C_{1,i-1}^HR_{1,i-1}(1)D_{1,i}^H,S_{i}(1)\rangle +\sum ^\theta _{i=1}\langle A_{2,i}^HR_{2,i}(1)B_{2,i}^H,S_i(1)\rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\langle C_{2,i-1}^HR_{2,i-1}(1)D_{2,i-1}^H,S_{i}(1)\rangle \Bigg ]\nonumber \\&\quad =\sum ^\theta _{i=1}\left[ ||R_{1,i}(1)||+||R_{2,i}(1)||\right] -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2 \right] }{\sum ^\theta _{j=1}||S_j(1)||^2}\nonumber \\&\qquad \times \Bigg [ \sum ^\theta _{i=1}\langle A_{1,i}^HR_{1,i}(1)B_{1,i}^H+C_{1,i-1}^HR_{1,i-1}(1)D_{1,i}^H+A_{2,i}^HR_{2,i}(1)B_{2,i}^H\nonumber \\&\qquad +\, C_{2,i-1}^HR_{2,i-1}(1)D_{2,i-1}^H,S_i(1)\rangle \Bigg ]=\sum ^\theta _{i=1}\left[ ||R_{1,i}(1)||+||R_{2,i}(1)||\right] \nonumber \\&\qquad -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }{\sum ^\theta _{j=1}||S_j(1)||^2} \sum ^\theta _{i=1}||S_i(1)||^2=0, \end{aligned}$$
(5.3)
and
$$\begin{aligned}&\!\!\!\!\!\!\!\sum ^\theta _{i=1}\Big \langle S_i(1), S_i(2)\Big \rangle =\sum ^\theta _{i=1}\Big \langle S_i(1), A_{1,i}^HR_{1,i}(2)B_{1,i}^H+C_{1,i-1}^HR_{1,i-1}(2)D_{1,i-1}^H\nonumber \\&\!\!\!\!\!\!\!\qquad +\, A_{2,i}^HR_{2,i}(2)B_{2,i}^H+C_{2,i-1}^HR_{2,i-1}(2)D_{2,i-1}^H\Big \rangle \nonumber \\&\!\!\!\!\!\!\!\qquad +\,\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(2)||^2+||R_{2,j}(2)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }\sum ^\theta _{i=1}\Big \langle S_i(1),S_i(1)\Big \rangle \nonumber \\&\!\!\!\!\!\!\!\quad =\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(2)||^2+||R_{2,j}(2)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2 \right] }\sum ^\theta _{i=1}|| S_i(1)||^2 \nonumber \\&\!\!\!\!\!\!\!\qquad +\,\sum ^\theta _{i=1}\Big \langle A_{1,i}S_i(1)B_{1,i}, R_{1,i}(2) \Big \rangle \nonumber \\&\!\!\!\!\!\!\!\qquad +\,\sum ^\theta _{i=1}\Big \langle C_{1,i}S_{i+1}(1)D_{1,i}, R_{1,i}(2)\Big \rangle +\sum ^\theta _{i=1}\Big \langle A_{2,i}S_i(1)B_{2,i}, R_{2,i}(2) \Big \rangle \nonumber \\&\!\!\!\!\!\!\!\qquad +\, \sum ^\theta _{i=1}\Big \langle C_{2,i}S_{i+1}(1)D_{2,i}, R_{2,i}(2) \Big \rangle \nonumber \\&\!\!\!\!\!\!\!\quad =\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(2)||^2+||R_{2,j}(2)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }\sum ^\theta _{i=1}|| S_i(1)||^2\nonumber \\&\!\!\!\!\!\!\!\qquad +\, \sum ^\theta _{i=1}\Big \langle A_{1,i}S_i(1)B_{1,i}+C_{1,i}S_{i+1}(1)D_{1,i}, R_{1,i}(2)\Big \rangle \nonumber \\&\!\!\!\!\!\!\!\qquad +\, \sum ^\theta _{i=1}\Big \langle A_{2,i}S_i(1)B_{2,i}+C_{2,i}S_{i+1}(1)D_{2,i}, R_{2,i}(2)\Big \rangle \nonumber \\&\!\!\!\!\!\!\!\quad = \frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(2)||^2+||R_{2,j}(2)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }\sum ^\theta _{i=1}|| S_i(1)||^2\nonumber \\&\!\!\!\!\!\!\!\qquad +\,\frac{\sum ^\theta _{j=1}||S_j(1)||^2}{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2\right] }\Bigg [\sum ^\theta _{i=1}\Big \langle R_{1,i}(1)-R_{1,i}(2), R_{1,i}(2)\Big \rangle \nonumber \\&\!\!\!\!\!\!\!\qquad +\, \sum ^\theta _{i=1}\Big \langle R_{2,i}(1)-R_{2,i}(2), R_{2,i}(2) \Big \rangle \Bigg ]\nonumber \\&\!\!\!\!\!\!\!\quad = \frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(2)||^2+||R_{2,j}(2)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2+||R_{2,j}(1)||^2 \right] }\sum ^\theta _{i=1}|| S_i(1)||^2\nonumber \\&\!\!\!\!\!\!\!\qquad -\,\frac{\sum ^\theta _{j=1}||S_j(1)||^2}{\sum ^\theta _{j=1}\left[ ||R_{1,j}(1)||^2\!+\!||R_{2,j}(1)||^2\right] }\sum ^\theta _{i=1}\left[ ||R_{1,i}(2)||^2 +||R_{2,i}(2)||^2\right] =0.\qquad \!\!\!\quad \end{aligned}$$
(5.4)
Step II In this step, for \(u<w<r\) we assume that
$$\begin{aligned} \sum ^\theta _{i=1}\left[ \Big \langle R_{1,i}(u), R_{1,i}(w)\Big \rangle +\Big \langle R_{2,i}(u), R_{2,i}(w)\Big \rangle \right] =0,~~~~\mathrm {and}~~~~~ \sum ^\theta _{i=1}\langle S_i(u), S_i(w)\rangle =0. \end{aligned}$$
We have
$$\begin{aligned}&\sum ^\theta _{i=1}\left[ \Big \langle R_{1,i}(u), R_{1,i}(w+1)\Big \rangle +\Big \langle R_{2,i}(u), R_{2,i}(w+1)\Big \rangle \right] \nonumber \\&\quad =\sum ^\theta _{i=1}\langle R_{1,i}(u), R_{1,i}(w)-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2}\nonumber \\&\qquad \times [ A_{1,i}S_i(w)B_{1,i}+C_{1,i}S_{i+1}(w)D_{1,i}]\rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\langle R_{2,i}(u), R_{2,i}(w)-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2}\nonumber \\&\qquad \times \left[ A_{2,i}S_i(w)B_{2,i}+C_{2,i}S_{i+1}(w)D_{2,i}\right] \rangle \nonumber \\&\quad =-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2}\left[ \sum ^\theta _{i=1}\Big \langle A_{1,i}^HR_{1,i}(u)B_{1,i}^H, S_i(w)\Big \rangle \right. \nonumber \\&\left. \qquad +\sum ^\theta _{i=1}\Big \langle C_{1,i-1}^HR_{1,i-1}(u)D_{1,i-1}^H,S_{i}(w)\Big \rangle \right. \nonumber \\&\left. \qquad + \sum ^\theta _{i=1}\Big \langle A_{2,i}^HR_{2,i}(u)B_{2,i}^H, S_i(w)\Big \rangle + \sum ^\theta _{i=1}\Big \langle C_{2,i-1}^HR_{2,i-1}(u)D_{2,i-1}^H,S_{i}(w)\Big \rangle \right] \nonumber \\&\quad =-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2}\nonumber \\&\qquad \times \left[ \sum ^\theta _{i=1}\Big \langle A_{1,i}^HR_{1,i}(u)B_{1,i}^H+C_{1,i-1}^HR_{1,i-1}(u)D_{1,i-1}^H+A_{2,i}^HR_{2,i}(u)B_{2,i}^H \right. \nonumber \\&\left. \qquad +\,C_{2,i-1}^HR_{2,i-1}(u)D_{2,i-1}^H, S_i(w)\Big \rangle \right] \nonumber \\&\quad =-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2}\left[ \sum ^\theta _{i=1}\Big \langle S_i(u)\right. \nonumber \\&\left. \qquad - \frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(u)||^2+||R_{2,j}(u)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(u-1)||^2+||R_{2,j}(u-1)||^2\right] }S_i(u-1), S_i(w)\Big \rangle \right] =0,\nonumber \\ \end{aligned}$$
(5.5)
and
$$\begin{aligned}&\sum ^\theta _{i=1}\Big \langle S_i(u), S_i(w+1)\Big \rangle = \sum ^\theta _{i=1}\Big \langle S_i(u), A_{1,i}^HR_{1,i}(w+1)B_{1,i}^H+C_{1,i-1}^HR_{1,i-1}(w+1)D_{1,i-1}^H\nonumber \\&\qquad +\, A_{2,i}^HR_{2,i}(w+1)B_{2,i}^H+C_{2,i-1}^HR_{2,i-1}(w+1)D_{2,i-1}^H\Big \rangle \nonumber \\&\qquad +\, \frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w+1)||^2+||R_{2,j}(w+1)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] } \sum ^\theta _{i=1}\Big \langle S_i(u),S_i(w)\Big \rangle \nonumber \\&\quad = \sum ^\theta _{i=1}\Big \langle A_{1,i}S_i(u)B_{1,i}, R_{1,i}(w+1) \Big \rangle +\sum ^\theta _{i=1}\Big \langle C_{1,i}S_{i+1}(u)D_{1,i}, R_{1,i}(w+1)\Big \rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\Big \langle A_{2,i}S_i(u)B_{2,i}, R_{2,i}(w+1) \Big \rangle + \sum ^\theta _{i=1}\Big \langle C_{2,i}S_{i+1}(u)D_{2,i}, R_{2,i}(w+1) \Big \rangle \nonumber \\&\quad = \frac{\sum ^\theta _{j=1}||S_j(u)||^2}{\sum ^\theta _{j=1}\left[ ||R_{1,j}(u)||^2+||R_{2,j}(u)||^2\right] }\left[ \sum ^\theta _{i=1}\Big \langle R_{1,i}(u)-R_{1,i}(u+1), R_{1,i}(w+1) \Big \rangle \right. \nonumber \\&\left. \qquad + \sum ^\theta _{i=1}\Big \langle R_{2,i}(u)-R_{2,i}(u+1), R_{2,i}(w+1) \Big \rangle \right] =0. \end{aligned}$$
(5.6)
For \(u = w\), we can get
$$\begin{aligned}&\sum ^\theta _{i=1}\left[ \Big \langle R_{1,i}(w), R_{1,i}(w+1)\Big \rangle +\Big \langle R_{2,i}(w), R_{2,i}(w+1)\Big \rangle \right] \nonumber \\&\quad =\sum ^\theta _{i=1}\Big \langle R_{1,i}(w), R_{1,i}(w)-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2}\nonumber \\&\qquad \times \left[ A_{1,i}S_i(w)B_{1,i}+C_{1,i}S_{i+1}(w)D_{1,i}\right] \Big \rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\Big \langle R_{2,i}(w), R_{2,i}(w)-\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2}\nonumber \\&\qquad \times \,[ A_{2,i}S_i(w)B_{2,i}+C_{2,i}S_{i+1}(w)D_{2,i}]\Big \rangle \nonumber \\&\quad = \sum ^\theta _{i=1}\left[ ||R_{1,i}(w)||^2+||R_{2,i}(w)||^2\right] -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2}\nonumber \\&\qquad \times \left[ \sum ^\theta _{i=1}\Big \langle A_{1,i}^HR_{1,i}(w)B_{1,i}^H+C_{1,i-1}^HR_{1,i-1}(w)D_{1,i-1}^H\right. \nonumber \\&\left. \qquad +A_{2,i}^HR_{2,i}(w)B_{2,i}^H+C_{2,i-1}^HR_{2,i-1}(w)D_{2,i-1}^H, S_i(w)\Big \rangle \right] \nonumber \\&\quad = \sum ^\theta _{i=1}\left[ ||R_{1,i}(w)||^2+||R_{2,i}(w)||^2\right] \nonumber \\&\qquad -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2} \sum ^\theta _{i=1}\Big \langle S_i(w)\nonumber \\&\qquad -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w-1)||^2+||R_{2,j}(w-1)||^2\right] }S_i(w-1), S_i(w)\Big \rangle \nonumber \\&\quad =\sum ^\theta _{i=1}\left[ ||R_{1,i}(w)||^2+||R_{2,i}(w)||^2\right] \nonumber \\&\qquad -\frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }{\sum ^\theta _{j=1}||S_j(w)||^2} \sum ^\theta _{i=1}||S_i(w)||^2=0, \end{aligned}$$
(5.7)
$$\begin{aligned}&\sum ^\theta _{i=1}\Big \langle S_i(w), S_i(w+1)\Big \rangle \nonumber \\&\quad = \sum ^\theta _{i=1}\Big \langle S_i(w), A_{1,i}^HR_{1,i}(w+1)B_{1,i}^H+C_{1,i-1}^HR_{1,i-1}(w+1)D_{1,i-1}^H\nonumber \\&\qquad + A_{2,i}^HR_{2,i}(w+1)B_{2,i}^H+C_{2,i-1}^HR_{2,i-1}(w+1)D_{2,i-1}^H\Big \rangle \nonumber \\&\qquad + \frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w+1)||^2+||R_{2,j}(w+1)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] } \sum ^\theta _{i=1}\Big \langle S_i(w),S_i(w)\Big \rangle \nonumber \\&\quad = \frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w+1)||^2+||R_{2,j}(w+1)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] } \sum ^\theta _{i=1}||S_i(w)||^2\nonumber \\&\qquad + \sum ^\theta _{i=1}\Big \langle A_{1,i}S_i(w)B_{1,i}, R_{1,i}(w+1) \Big \rangle +\sum ^\theta _{i=1}\Big \langle C_{1,i}S_{i+1}(w)D_{1,i}, R_{1,i}(w+1)\Big \rangle \nonumber \\&\qquad +\sum ^\theta _{i=1}\Big \langle A_{2,i}S_i(w)B_{2,i}, R_{2,i}(w+1) \Big \rangle + \sum ^\theta _{i=1}\Big \langle C_{2,i}S_{i+1}(w)D_{2,i}, R_{2,i}(w+1) \Big \rangle \nonumber \\&\quad = \frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w+1)||^2+||R_{2,j}(w+1)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] } \sum ^\theta _{i=1}||S_i(w)||^2\nonumber \\&\qquad + \frac{\sum ^\theta _{j=1}||S_j(w)||^2}{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }\left[ \sum ^\theta _{i=1}\Big \langle R_{1,i}(w)-R_{1,i}(w+1), R_{1,i}(w+1) \Big \rangle \right. \nonumber \\&\left. \qquad + \sum ^\theta _{i=1}\Big \langle R_{2,i}(w)-R_{2,i}(w+1), R_{2,i}(w+1) \Big \rangle \right] \nonumber \\&\quad = \frac{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w+1)||^2+||R_{2,j}(w+1)||^2\right] }{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] } \sum ^\theta _{i=1}||S_i(w)||^2\nonumber \\&\qquad - \frac{\sum ^\theta _{j=1}||S_j(w)||^2}{\sum ^\theta _{j=1}\left[ ||R_{1,j}(w)||^2+||R_{2,j}(w)||^2\right] }\sum ^\theta _{i=1}\left[ ||R_{1,i}(w+1)||^2+||R_{2,i}(w+1)||^2\right] =0. \nonumber \\ \end{aligned}$$
(5.8)
By considering Steps I and II, two statements of Lemma 2 hold by the principle of induction.
