## Appendix B --Proof of (34) and (35)  
Applying Proposition 1 and (31) leads to  
\[\begin{array}{ll}\hat{J}(\hat{p})&=&\lambda_{\max}\{\hat{B}^{\rm H}(\hat{p}) \hat{\bar{B}}(\hat{p})\}=\lambda_{\max}\{\mathcal{C}_{\rm Q}\}+\mathbf{u}_{N}^{\rm H }(\mathcal{C}_{\rm O}+\mathbf{C}^{(1)}+\mathbf{C}^{(2)}+o(||\widetilde{\mathbf{\xi}}||_{2} ^{2}))\mathbf{u}_{N}\\ &&+\mathbf{u}_{N}^{\rm H}(\mathcal{C}_{\rm O}+\mathbf{C}^{(1)}+\mathbf{C}^{(2)}+o(|| \widetilde{\mathbf{\xi}}||_{2}^{2}))\mathbf{u}_{N}(\mathcal{C}_{\rm O}+\mathbf{C}^{(1)}+ \mathbf{C}^{(2)}+o(||\widetilde{\mathbf{\xi}}||_{2}^{2}))\mathbf{u}_{N}+o(||\mathbf{C}^{(1)}+ \mathbf{C}^{(2)}+o(||\widetilde{\mathbf{\xi}}||_{2}^{2})||_{2}^{2})\\ &=&\lambda_{\max}\{\mathcal{C}_{\rm O}\}+\mathbf{u}_{N}^{\rm H}(\mathbf{C}^{(1)}+\mathbf{ C}^{(2)})\mathbf{u}_{N}+\mathbf{u}_{N}^{\rm H}\mathbf{C}^{(1)}\mathbf{u}_{N}\mathbf{C}^{(1)}\mathbf{u}_{N}+o(|| \widetilde{\mathbf{\xi}}||_{2}^{2})\\ &=&\lambda_{N}+\widetilde{\lambda}_{N}^{(1)}+\widetilde{\lambda}_{N}^{(2)}+o(|| \widetilde{\mathbf{\xi}}||_{2}^{2})\,,\end{array} \tag{100}\]  
where \(\widetilde{\lambda}_{N}^{(1)}\) and \(\widetilde{\lambda}_{N}^{(2)}\) consist of all first- and second-order error terms, respectively. It then follows that  
\[\left\{\begin{array}{l}\widetilde{\lambda}_{N}^{(1)}=\mathbf{u}_{N}^{\rm H}\mathbf{ C}^{(1)}\mathbf{u}_{N}=\mathcal{O}(||\widetilde{\mathbf{\xi}}||_{2})\,,\\ \widetilde{\lambda}_{N}^{(2)}=\mathbf{u}_{N}^{\rm H}\mathbf{C}^{(2)}\mathbf{u}_{N}+\mathbf{u}_ {N}^{\rm H}\mathbf{C}^{(1)}\mathbf{U}_{N}\mathbf{C}^{(1)}\mathbf{u}_{N}=\mathcal{O}(|| \widetilde{\mathbf{\xi}}||_{2}^{2})\,.\end{array}\right. \tag{101}\]  
Inserting the second equality in (32) into the first equality in (101) leads to  
\[\widetilde{\lambda}_{N}^{(1)}=\mathbf{u}_{N}^{\rm H}(\mathbf{B}_{0}^{\rm H}\widetilde {\mathbf{B}}^{(1)}+\widetilde{\mathbf{B}}^{(1){\rm H}}\mathbf{B}_{0})\mathbf{u}_{N}=\mathbf{u}_{N} ^{\rm H}\mathbf{B}_{0}^{\rm H}\widetilde{\mathbf{B}}^{(1)}\mathbf{u}_{N}+\left(\mathbf{u}_{N}^ {\rm H}\mathbf{B}_{0}^{\rm H}\widetilde{\mathbf{B}}^{(1)}\mathbf{u}_{N}\right)^{\rm H}= \mathcal{O}(||\widetilde{\mathbf{\xi}}||_{2})\,. \tag{102}\]  
Substituting the second and third equalities in (32) into the second equality in (101) leads to  
\[\begin{array}{l}\widetilde{\lambda}_{N}^{(2)}=\mathbf{u}_{N}^{\rm H}(\widetilde {\mathbf{B}}^{(1){\rm H}}\widetilde{\mathbf{B}}^{(1)}+\mathbf{B}_{0}^{\rm H}\widetilde\mathbf{{B}}^{(2)}+\widetilde{\mathbf{B}}^{(2){\rm H}}\mathbf{B}_{0})\mathbf{u}_{N}+\mathbf{u}_{N}^{\rm H }(\mathbf{B}_{0}^{\rm H}\widetilde{\mathbf{B}}^{(1)}+\widetilde{\mathbf{B}}^{(1){\rm H}}\bm {B}_{0})\mathbf{u}_{N}\\ \phantom{=}=\mathbf{u}_{N}^{\rm H}\mathbf{B}_{0}^{\rm H}\widetilde{\mathbf{B}}^{(1)}\mathbf{u}_ {N}\mathbf{B}_{0}^{\rm H}\widetilde{\mathbf{B}}^{(1)}\mathbf{u}_{N}+(\mathbf{u}_{N}^{\rm H} \mathbf{B}_{0}^{\rm H}\widetilde{\mathbf{B}}^{(1)}\mathbf{u}_{N}\mathbf{B}_{0}^{\rm H} \widetilde{\mathbf{B}}^{(1)}\mathbf{u}_{N})^{\rm H}+\mathbf{u}_{N}^{\rm H}\widetilde{\mathbf{ B}}^{(1){\rm H}}(\mathbf{I}_{K}+\mathbf{B}_{0}\mathbf{U}_{N}\mathbf{B}_{0}^{\rm H}) \widetilde{\mathbf{B}}^{(1)}\mathbf{u}_{N}\\ \phantom{=}+\mathbf{u}_{N}^{\rm H}\mathbf{B}_{0}^{\rm H}\widetilde{\mathbf{B}}^{(1)}\mathbf{U}_ {N}\widetilde{\mathbf{B}}^{(1){\rm H}}\mathbf{B}_{0}\mathbf{u}_{N}+\mathbf{u}_{N}^{\rm H}\mathbf{B }_{0}^{\rm H}\widetilde{\mathbf{B}}^{(2)}\mathbf{u}_{N}+(\mathbf{u}_{N}^{\rm H}\mathbf{B}_{0}^{ \rm H}\widetilde{\mathbf{B}}^{(2)}\mathbf{u}_{N})^{\rm H}=\mathcal{O}(||\widetilde{ \mathbf{\xi}}||_{2}^{2})\,.\end{array} \tag{103}\]  
Combining (102) and (103) completes the proof.